A097750 -id:A097750 - cp's OEIS frontend

A097761 Inverse of binomial transform of Whitney triangle.

1, -2, 1, 4, -4, 1, -10, 13, -6, 1, 28, -42, 26, -8, 1, -84, 138, -102, 43, -10, 1, 264, -462, 385, -198, 64, -12, 1, -858, 1573, -1430, 845, -338, 89, -14, 1, 2860, -5434, 5278, -3458, 1610, -530, 118, -16, 1, -9724, 19006, -19448, 13804, -7208, 2788, -782, 151, -18, 1, 33592

Offset: 0

Paul Barry, Aug 23 2004

As an element of the Riordan group, this is ((1-x*C(-x))*C(-x), x*C(-x)^2), where C(x) = (1-(1-4*x)^(1/2))/(2*x) is the g.f. for Catalan numbers, A000108.
Inverse of A097750.
Row sums have g.f. (1-x*C(-x))*C(-x)/(1-x*C(-x)^2), or 1,-1,1,-2,5,-14,42,...

			Rows begin:
  {1},
  {-2,1},
  {4,-4,1},
  {-10,13,-6,1},
  ...

A131250 A007318 * A004070.

Original entry on oeis.org

1, 2, 1, 4, 4, 1, 8, 11, 6, 1, 16, 26, 22, 8, 1, 32, 57, 64, 37, 10, 1, 64, 120, 163, 130, 56, 12, 1, 128, 247, 382, 386, 232, 79, 14, 1, 256, 502, 848, 1024, 794, 378, 106, 16, 1, 512, 1013, 1816, 2510, 2380, 1471, 576, 137, 18, 1

Offset: 0

Gary W. Adamson, Jun 23 2007

Row sums = A061667: (1, 3, 9, 26, 73, 201, ...).
Companion triangle = A131249 = A007318 * A052509, where A052509 is the reversal of A004070.
Reversal of A097750. - Philippe Deléham, Jan 11 2014
Riordan array (1/(1-2x), x/(1-x)^2). - Philippe Deléham, Jan 11 2014
Diagonal sums are A045623. - Philippe Deléham, Jan 11 2014

			First few rows of the triangle:
   1;
   2,  1;
   4,  4,  1;
   8, 11,  6,  1;
  16, 26, 22,  8,  1;
  32, 57, 64, 37, 10,  1;
  ...

Filippo Disanto, Some Statistics on the Hypercubes of Catalan Permutations, Journal of Integer Sequences, Vol. 18 (2015), Article 15.2.2 (see Table 2).

Cf. A004070, A061667, A131249, A007318.

Binomial transform of A004070.

T(n,k) = 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k), T(0,0)=1, T(n,k)=0 if k < 0 or if k > n. - Philippe Deléham, Jan 11 2014

More terms from Philippe Deléham, Jan 11 2014

A320904 T(n, k) = binomial(2n + 1 - k, k)hypergeom([1, 1, -k], [1, 2*(n - k + 1)], -1), triangle read by rows, T(n, k) for n >= 0 and 0 <= k <= n.

Original entry on oeis.org

1, 1, 3, 1, 5, 7, 1, 7, 16, 15, 1, 9, 29, 42, 31, 1, 11, 46, 93, 99, 63, 1, 13, 67, 176, 256, 219, 127, 1, 15, 92, 299, 562, 638, 466, 255, 1, 17, 121, 470, 1093, 1586, 1486, 968, 511, 1, 19, 154, 697, 1941, 3473, 4096, 3302, 1981, 1023

Offset: 0

Peter Luschny, Oct 28 2018

			Triangle starts:
[0] 1
[1] 1,  3
[2] 1,  5,   7
[3] 1,  7,  16,  15
[4] 1,  9,  29,  42,   31
[5] 1, 11,  46,  93,   99,   63
[6] 1, 13,  67, 176,  256,  219,  127
[7] 1, 15,  92, 299,  562,  638,  466, 255
[8] 1, 17, 121, 470, 1093, 1586, 1486, 968, 511

Row sums are A105693(n-1).

Cf. A097750.

T := (n, k) -> binomial(2*n + 1 - k, k)*hypergeom([1, 1, -k], [1, 2*(n-k+1)], -1):
for n from 0 to 11 do seq(simplify(T(n, k)), k = 0..n) od;

s={};For[n=0,n<19,n++,For[k=0,kDetlef Meya, Oct 03 2023 *)

cp's OEIS Frontend

A097761 Inverse of binomial transform of Whitney triangle.

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A131250 A007318 * A004070.

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A320904 T(n, k) = binomial(2n + 1 - k, k)hypergeom([1, 1, -k], [1, 2*(n - k + 1)], -1), triangle read by rows, T(n, k) for n >= 0 and 0 <= k <= n.

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cp's OEIS Frontend

A097761 Inverse of binomial transform of Whitney triangle.

Views

Author

Keywords

Comments

Examples

A131250 A007318 * A004070.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

Extensions

A320904 T(n, k) = binomial(2*n + 1 - k, k)*hypergeom([1, 1, -k], [1, 2*(n - k + 1)], -1), triangle read by rows, T(n, k) for n >= 0 and 0 <= k <= n.

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Mathematica

A320904 T(n, k) = binomial(2n + 1 - k, k)hypergeom([1, 1, -k], [1, 2*(n - k + 1)], -1), triangle read by rows, T(n, k) for n >= 0 and 0 <= k <= n.