Worpitzky's Identity Extended

BBB gives the recognized Worpitzky Identity of 1883.  It defines an x^n value.

AAA, (and PPP and QQQ) are examples of shell/nexus number series.  At any lower level of recursive accumulation than the level of nexus numbers to powers, the algorithm is imperfect for first values in series (but not most of the values in series).

CCC, DDD and EEE show the algorithm extending to summations of summations of series (perfectly).

In[1]:=

<<DiscreteMath`Combinatorica`

In[2]:=

AAA = Underoverscript[∑, k = 0, arg3] Eulerian[n, k] Binomial[-k + n + x - 1, n - 1]

In[3]:=

Table[AAA,{n,5},{x,0,10}]//MatrixForm

Out[3]//MatrixForm=

( {{1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}, {1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21}, {1, 7 ... 5, 2465, 3439, 4641}, {1, 31, 211, 781, 2101, 4651, 9031, 15961, 26281, 40951, 61051}} )

In[4]:=

BBB = Underoverscript[∑, k = 0, arg3] Eulerian[n, k] Binomial[-k + n + x, n]

In[5]:=

Table[BBB, {n, 5}, {x, 0, 10}]//MatrixForm

Out[5]//MatrixForm=

( {{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}, {1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121}, ... 1, 10000, 14641}, {1, 32, 243, 1024, 3125, 7776, 16807, 32768, 59049, 100000, 161051}} )

In[6]:=

CCC = Underoverscript[∑, k = 0, arg3] Eulerian[n, k] Binomial[-k + n + x + 1, n + 1]

In[7]:=

Table[CCC, {n, 5}, {x, 0, 10}]//MatrixForm

Out[7]//MatrixForm=

( {{1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66}, {1, 5, 14, 30, 55, 91, 140, 204, 285, ... 25333, 39974}, {1, 33, 276, 1300, 4425, 12201, 29008, 61776, 120825, 220825, 381876}} )

In[8]:=

DDD = Underoverscript[∑, k = 0, arg3] Eulerian[n, k] Binomial[-k + n + x + 2, n + 2]

In[9]:=

Table[DDD, {n, 5}, {x, 0, 10}]//MatrixForm

Out[9]//MatrixForm=

( {{1, 4, 10, 20, 35, 56, 84, 120, 165, 220, 286}, {1, 6, 20, 50, 105, 196, 336, 540 ... 57838, 97812}, {1, 34, 310, 1610, 6035, 18236, 47244, 109020, 229845, 450670, 832546}} )

In[10]:=

EEE = Underoverscript[∑, k = 0, arg3] Binomial[-k + n + x + 3, n + 3] Eulerian[n, k]

In[11]:=

Table[EEE, {n, 5}, {x, 0, 10}]//MatrixForm

Out[11]//MatrixForm=

( {{1, 5, 15, 35, 70, 126, 210, 330, 495, 715, 1001}, {1, 7, 27, 77, 182, 378, 714, ... 693, 219505}, {1, 35, 345, 1955, 7990, 26226, 73470, 182490, 412335, 863005, 1695551}} )

In[12]:=

PPP = Underoverscript[∑, k = 0, arg3] Eulerian[n, k] Binomial[-k + n + x - 2, n - 2]

In[13]:=

Table[PPP, {n, 5}, {x, 0, 10}]//MatrixForm

Out[13]//MatrixForm=

( {{0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2}, {0, 6, 12, ... 590, 770, 974, 1202}, {0, 30, 180, 570, 1320, 2550, 4380, 6930, 10320, 14670, 20100}} )

In[14]:=

QQQ = Underoverscript[∑, k = 0, arg3] Eulerian[n, k] Binomial[-k + n + x - 3, n - 3]

In[15]:=

Table[QQQ, {n, 5}, {x, 0, 10}]//MatrixForm

Out[15]//MatrixForm=

( {{0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {6, 6, 6, 6 ... 132, 156, 180, 204, 228}, {30, 30, 150, 390, 750, 1230, 1830, 2550, 3390, 4350, 5430}} )

Created by Mathematica  (April 13, 2005) Valid XHTML 1.1!