The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A326326 (Bold, blue-underlined text is an addition; faded, red-underlined text is a ~~deletion~~.)
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A326326

T(n, k) = [x^k] Sum_{j=0..n} Pochhammer(x, j), for 0 <= k <= n, triangle read by rows.
(history; published version)

#11 by Peter Luschny at Mon Apr 11 16:18:41 EDT 2022

STATUS

editing

approved

#10 by Peter Luschny at Mon Apr 11 16:18:06 EDT 2022

FORMULA

Conjecture: T(n,k) = Sum_{i=0..n} A132393(i,k) for 0 <= k <= n (conjectured). - Werner Schulte, Mar 30 2022

STATUS

proposed

editing

Discussion

Mon Apr 11

16:18

Peter Luschny: The conjecture is true.

#9 by Werner Schulte at Wed Mar 30 15:58:17 EDT 2022

STATUS

editing

proposed

#8 by Werner Schulte at Wed Mar 30 15:56:28 EDT 2022

FORMULA

T(n,k) = Sum_{i=0..n} A132393(i,k) for 0 <= k <= n (conjectured). - Werner Schulte, Mar 30 2022

CROSSREFS

Cf. A265609, A132393.

STATUS

approved

editing

#7 by Peter Luschny at Tue Jul 02 14:18:27 EDT 2019

STATUS

editing

approved

#6 by Peter Luschny at Tue Jul 02 14:12:15 EDT 2019

MATHEMATICA

Table[CoefficientList[FunctionExpand[Sum[Pochhammer[x, k], {k, 0, n}]], x], {n, 0, 10}] // Flatten

CROSSREFS

The inverse of the lower triangular matrix is the signed form of A256894.

#5 by Peter Luschny at Tue Jul 02 12:56:43 EDT 2019

CROSSREFS

First Second column is A003422(n) and row sums are A003422(n+1).

Third column is A097422.

Cf. A265609.

#4 by Peter Luschny at Tue Jul 02 12:26:39 EDT 2019

CROSSREFS

Same construct construction for the falling factorial is A176663.

First column is A003422(n) and row sums are A003422(n+1).

Alternating row sums are A000007.

#3 by Peter Luschny at Tue Jul 02 12:12:44 EDT 2019

EXAMPLE

Triangle starts:

[0] [1]

[1] [1, 1]

[2] [1, 2, 1]

[3] [1, 4, 4, 1]

[4] [1, 10, 15, 7, 1]

[5] [1, 34, 65, 42, 11, 1]

[6] [1, 154, 339, 267, 96, 16, 1]

[7] [1, 874, 2103, 1891, 831, 191, 22, 1]

[8] [1, 5914, 15171, 15023, 7600, 2151, 344, 29, 1]

[9] [1, 46234, 124755, 133147, 74884, 24600, 4880, 575, 37, 1]

#2 by Peter Luschny at Tue Jul 02 12:09:35 EDT 2019

NAME

allocated T(n, k) = [x^k] Sum_{j=0..n} Pochhammer(x, j), for Peter Luschny0 <= k <= n, triangle read by rows.

DATA

1, 1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 10, 15, 7, 1, 1, 34, 65, 42, 11, 1, 1, 154, 339, 267, 96, 16, 1, 1, 874, 2103, 1891, 831, 191, 22, 1, 1, 5914, 15171, 15023, 7600, 2151, 344, 29, 1, 1, 46234, 124755, 133147, 74884, 24600, 4880, 575, 37, 1

OFFSET

0,5

FORMULA

Sum_{k=0..n) T(n, k)*x^k = Sum_{k=0..n) (x)^k, where (x)^k denotes the rising factorial.

MAPLE

with(PolynomialTools):

T_row := n -> CoefficientList(expand(add(pochhammer(x, j), j=0..n)), x):

ListTools:-Flatten([seq(T_row(n), n=0..9)]);

CROSSREFS

Same construct for the falling factorial is A176663.

KEYWORD

allocated

nonn,tabl

AUTHOR

Peter Luschny, Jul 02 2019

STATUS

approved

editing