A140356 -id:A140356 - cp's OEIS frontend

A167172 Triangle T(n,k) read by rows: T(n,k) = binomial(n, k) + A140356(n, k) - 1.

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 7, 4, 1, 1, 5, 11, 11, 5, 1, 1, 6, 16, 25, 16, 6, 1, 1, 7, 22, 40, 40, 22, 7, 1, 1, 8, 29, 61, 93, 61, 29, 8, 1, 1, 9, 37, 89, 149, 149, 89, 37, 9, 1, 1, 10, 46, 125, 233, 371, 233, 125, 46, 10, 1

Offset: 0

Roger L. Bagula, Oct 29 2009

Row sums are: {1, 2, 4, 8, 17, 34, 71, 140, 291, 570, 1201,...}.

			{1},
{1, 1},
{1, 2, 1},
{1, 3, 3, 1},
{1, 4, 7, 4, 1},
{1, 5, 11, 11, 5, 1},
{1, 6, 16, 25, 16, 6, 1},
{1, 7, 22, 40, 40, 22, 7, 1},
{1, 8, 29, 61, 93, 61, 29, 8, 1},
{1, 9, 37, 89, 149, 149, 89, 37, 9, 1},
{1, 10, 46, 125, 233, 371, 233, 125, 46, 10, 1}.

G. C. Greubel, Table of n, a(n) for the first 50 rows

Cf. A140356.

t[n_, k_] = Binomial[n, k] + If[k <= Floor[n/2], Gamma[k + 1], Gamma[n - k + 1]] - 1; Flatten[Table[Table[t[n, k], {k, 0, n}], {n, 0, 10}]]

T(n,k) = binomial(n, k) + A140356(n, k) - 1 = binomial(n, k) + if(k less than equal floor(n/2), Gamma(k + 1), Gamma(n - k + 1)) - 1.

Edited by the OEIS editors, Jun 05 2016

A225413 Triangle read by rows: T(n,k) = (A101164(n,k) - A014473(n,k))/2.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 3, 3, 0, 0, 0, 0, 6, 12, 6, 0, 0, 0, 0, 10, 30, 30, 10, 0, 0, 0, 0, 15, 60, 91, 60, 15, 0, 0, 0, 0, 21, 105, 215, 215, 105, 21, 0, 0, 0, 0, 28, 168, 435, 590, 435, 168, 28, 0, 0, 0, 0, 36, 252, 791, 1365, 1365, 791, 252, 36, 0, 0

Offset: 0

Jeremy Gardiner, Jul 28 2013

Has opposite parity to A140356, A155454.

			Triangle begins as:
  0;
  0,  0;
  0,  0,  0;
  0,  0,  0,   0;
  0,  0,  1,   0,    0;
  0,  0,  3,   3,    0,    0;
  0,  0,  6,  12,    6,    0,    0;
  0,  0, 10,  30,   30,   10,    0,    0;
  0,  0, 15,  60,   91,   60,   15,    0,    0;
  0,  0, 21, 105,  215,  215,  105,   21,    0,    0;
  0,  0, 28, 168,  435,  590,  435,  168,   28,    0,   0;
  0,  0, 36, 252,  791, 1365, 1365,  791,  252,   36,   0,  0;
  0,  0, 45, 360, 1330, 2800, 3571, 2800, 1330,  360,  45,  0,  0;
  0,  0, 55, 495, 2106, 5250, 8197, 8197, 5250, 2106, 495, 55,  0,  0;

Reinhard Zumkeller, Rows n = 0..100 of table, flattened

Cf. A000129, A007318, A008288, A014473, A101164, A121262, A140356, A155454.
3rd column = A000217 (triangular numbers).
4th column = A027480 (n(n+1)(n+2)/2).

a225413 n k = a225413_tabl !! n !! k
a225413_row n = a225413_tabl !! n
a225413_tabl = map (map (`div` 2)) $
               zipWith (zipWith (-)) a101164_tabl a014473_tabl
-- Reinhard Zumkeller, Jul 30 2013

A008288:= func< n,k | (&+[Binomial(n-j, j)*Binomial(n-2*j, k-j): j in [0..k]]) >;
A225413:= func< n,k | (A008288(n,k) - 2*Binomial(n,k) + 1)/2 >;
[A225413(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Apr 08 2024

T[n_, k_]:= ((-1)^(n-k)*Hypergeometric2F1[-n+k,k+1,1,2] - 2*Binomial[n, k] +1)/2;
Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Apr 08 2024 *)

def A008288(n,k): return sum(binomial(n-j,j)*binomial(n-2*j,k-j) for j in range(k+1))
def A225413(n,k): return (A008288(n,k) -2*binomial(n,k) +1)//2
flatten([[A225413(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Apr 08 2024

T(n, k) = (A101164(n,k) - A014473(n,k))/2.
T(n, k) = (A008288(n,k) - 2*A007318(n,k) + 1)/2.
From G. C. Greubel, Apr 08 2024: (Start)
T(n, n-k) = T(n, k).
Sum_{k=0..n} T(n, k) = (A000129(n+1) + n + 1 - 2^(n+1))/2.
Sum_{k=0..n} (-1)^k*T(n, k) = A121262(n) - [n=0]. (End)

cp's OEIS Frontend

A167172 Triangle T(n,k) read by rows: T(n,k) = binomial(n, k) + A140356(n, k) - 1.

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