Euler's Triangle Extended

AAA gives Euler's Triangle values.

BBB through GGG are then formulas for the next six recursive summations of series from an initial Euler Triangle row.

The (n+1) th accumulation from an initial Euler Triangle row values gives a series that are x^n values.  The n th recursive summation gives shell/nexus number values.  The (n+2) th accumulation gives summations of 1st through x th of x^n.

The last formula summarizes these formulas and the "cube" of series.

In[1]:=

AAA = Underoverscript[∑, j = 0, arg3] (-1)^j * Binomial[n + 1, j] * (k - j + 1)^n

In[2]:=

Table[AAA,{n,7},{k,0,9}]//MatrixForm

Out[2]//MatrixForm=

( {{1, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {1, 1, 0, 0, 0, 0, 0, 0, 0, 0}, {1, 4, 1, 0, 0, 0 ... 0}, {1, 57, 302, 302, 57, 1, 0, 0, 0, 0}, {1, 120, 1191, 2416, 1191, 120, 1, 0, 0, 0}} )

In[3]:=

BBB = Underoverscript[∑, j = 0, arg3] (-1)^j * Binomial[n, j] * (k - j + 1)^n

In[4]:=

Table[BBB, {n, 7}, {k, 0, 9}]//MatrixForm

Out[4]//MatrixForm=

( {{1, 1, 1, 1, 1, 1, 1, 1, 1, 1}, {1, 2, 2, 2, 2, 2, 2, 2, 2, 2}, {1, 5, 6, 6, 6, 6 ... 9, 720, 720, 720, 720, 720}, {1, 121, 1312, 3728, 4919, 5039, 5040, 5040, 5040, 5040}} )

In[5]:=

CCC = Underoverscript[∑, j = 0, arg3] (-1)^j * Binomial[n - 1, j] * (k - j + 1)^n

In[6]:=

Table[CCC, {n, 7}, {k, 0, 9}]//MatrixForm

Out[6]//MatrixForm=

( {{1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, {1, 3, 5, 7, 9, 11, 13, 15, 17, 19}, {1, 6, 12, ... 40, 3960, 4680, 5400}, {1, 122, 1434, 5162, 10081, 15120, 20160, 25200, 30240, 35280}} )

In[7]:=

DDD = Underoverscript[∑, j = 0, arg3] (-1)^j * Binomial[n - 2, j] * (k - j + 1)^n

In[8]:=

Table[DDD, {n, 7}, {k, 0, 9}]//MatrixForm

Out[8]//MatrixForm=

( {{1, 3, 6, 10, 15, 21, 28, 36, 45, 55}, {1, 4, 9, 16, 25, 36, 49, 64, 81, 100}, {1 ... 3080, 17760, 23160}, {1, 123, 1557, 6719, 16800, 31920, 52080, 77280, 107520, 142800}} )

In[9]:=

EEE = Underoverscript[∑, j = 0, arg3] (-1)^j * Binomial[n - 3, j] * (k - j + 1)^n

In[10]:=

Table[EEE, {n, 7}, {k, 0, 9}]//MatrixForm

Out[10]//MatrixForm=

( {{1, 4, 10, 20, 35, 56, 84, 120, 165, 220}, {1, 5, 14, 30, 55, 91, 140, 204, 285, ... 40, 51300, 74460}, {1, 124, 1681, 8400, 25200, 57120, 109200, 186480, 294000, 436800}} )

In[11]:=

FFF = Underoverscript[∑, j = 0, arg3] (-1)^j * Binomial[n - 4, j] * (k - j + 1)^n

In[12]:=

Table[FFF, {n, 7}, {k, 0, 9}]//MatrixForm

Out[12]//MatrixForm=

( {{1, 5, 15, 35, 70, 126, 210, 330, 495, 715}, {1, 6, 20, 50, 105, 196, 336, 540, 8 ... 124802, 199262}, {1, 125, 1806, 10206, 35406, 92526, 201726, 388206, 682206, 1119006}} )

In[13]:=

GGG = Underoverscript[∑, j = 0, arg3] (-1)^j * Binomial[n - 5, j] * (k - j + 1)^n

In[14]:=

Table[GGG, {n, 7}, {k, 0, 9}]//MatrixForm

Out[14]//MatrixForm=

( {{1, 6, 21, 56, 126, 252, 462, 792, 1287, 2002}, {1, 7, 27, 77, 182, 378, 714, 125 ... 9297, 468559}, {1, 126, 1932, 12138, 47544, 140070, 341796, 730002, 1412208, 2531214}} )

MagicNKZ =   Underoverscript[∑, j = 0, arg3] (-1)^j ({{n - z + 1}, {j}}) ( ... 1;n is the powerk is the ordinal occurrencez is the level of recursive summation

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