A155495 -id:A155495 - cp's OEIS frontend

A288470 a(n) = Sum_{k=0..n} binomial(n,k)binomial(2n,2*k).

1, 2, 14, 92, 646, 4652, 34124, 253528, 1901638, 14368844, 109208164, 833981128, 6394017436, 49185717752, 379438594136, 2934361958192, 22741538394694, 176582855512588, 1373431963785332, 10698376362421096, 83447762846703796, 651690159076273192, 5095051571420324264, 39874449115469939152, 312350761370734541596

Offset: 0

Robert Israel, Jun 09 2017

Row sums of A155495.

a(n) is the constant term in the expansion of (-1 + (1 + x + 1/x)^2)^n. - Seiichi Manyama, Nov 21 2019

Seiichi Manyama, Table of n, a(n) for n = 0..1109

Cf. A094061, A155495, A218045.

f:= gfun:-rectoproc({n*(2*n-1)*a(n) = (32*(n-2))*(2*n-5)*a(n-3)+(8*(9*n^2-31*n+28))*a(n-2)+(2*(3*n^2+7*n-9))*a(n-1), a(0)=1,a(1)=2, a(2)=14},a(n),remember):
map(f, [$0..30]);

Table[Sum[Binomial[n, k] Binomial[2 n, 2 k], {k, 0, n}], {n, 0, 24}] (* Michael De Vlieger, Jun 09 2017 *)

{a(n) = polcoef((-1+(1+x+1/x)^2)^n, 0)} \\ Seiichi Manyama, Nov 21 2019

a(n) = hypergeom([-n,-n,1/2-n],[1/2,1],-1).
n*(2*n-1)*a(n) = (32*(n-2))*(2*n-5)*a(n-3)+(8*(9*n^2-31*n+28))*a(n-2)+(2*(3*n^2+7*n-9))*a(n-1).
G.f.: sqrt((1-2*x+sqrt(1-8*x))/(2*(1-7*x-8*x^2))).
a(n) ~ 8^n / sqrt(3*Pi*n). - Vaclav Kotesovec, Nov 27 2017
a(n) = Sum_{k=0..floor(n/2)} binomial(2*n,k) * binomial(3*n-2*k-1,n-2*k). - Seiichi Manyama, Feb 13 2024
From Peter Bala, Aug 30 2025: (Start)
n*(2*n - 1)*(3*n - 4)*a(n) = 2*(21*n^3 - 49*n^2 + 33*n - 6)*a(n-1) + 8*(n - 1)*(3*n - 1)*(2*n - 3)*a(n-2) with a(0) = 1 and a(1) = 2.
a(n) = Sum_{0 <= i, j <= n/2} binomial(2*n, j)*binomial(2*n+i-1, i)*binomial(2*n, n- 2*i-2*j) (verified to satisfy the above second-order recurrence using the MulZeil procedure in Doron Zeilberger's MultiZeilberger Maple package).
Equivalently, a(n) = [x^n] ( (1 + x + x^2 + x^3)/(1 - x^2) )^(2*n). Cf. A240688.
The Gauss congruences hold: a(n*p^k) == a(n*p^(k-1)) (mod p^k)  for all primes p and positive integers n and k.
Conjecture: the supercongruences a(n*p^k) == a(n*p^(k-1)) (mod p^(2*k)) hold for primes p >= 5 and positive integers n and k. (End)

A168518 Expansion of g.f. (1/2)( a(1+x)^n + b(1-x)^(n+2)LerchPhi(x, -n-1, 1) + c2^(n+1)(1-x)^(n+1)*LerchPhi(x, -n, 1/2) ), where a = -4, b = 2, and c = 2, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 12, 1, 1, 51, 51, 1, 1, 170, 514, 170, 1, 1, 521, 3646, 3646, 521, 1, 1, 1552, 22247, 49472, 22247, 1552, 1, 1, 4591, 125565, 534995, 534995, 125565, 4591, 1, 1, 13590, 677776, 5058698, 9506078, 5058698, 677776, 13590, 1, 1, 40341, 3560448, 43870968, 140136690, 140136690, 43870968, 3560448, 40341, 1

Offset: 0

Roger L. Bagula, Nov 28 2009

			Triangle begins as:
  1;
  1,     1;
  1,    12,       1;
  1,    51,      51,        1;
  1,   170,     514,      170,         1;
  1,   521,    3646,     3646,       521,         1;
  1,  1552,   22247,    49472,     22247,      1552,        1;
  1,  4591,  125565,   534995,    534995,    125565,     4591,       1;
  1, 13590,  677776,  5058698,   9506078,   5058698,   677776,   13590,     1;
  1, 40341, 3560448, 43870968, 140136690, 140136690, 43870968, 3560448, 40341, 1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A142460, A155491, A155495, A157273, A166343.

Cf. A168517, A168549, A168551, A168552.

p[x_, n_, a_, b_, c_]= (1/2)*(a*(1+x)^n + b*(1-x)^(n+2)*LerchPhi[x,-n-1,1] + c*2^(n+1)*(1-x)^(n+1)*LerchPhi[x,-n,1/2]);
Table[CoefficientList[p[x,n,-4,2,2], x], {n,0,10}]//Flatten (* modified by G. C. Greubel, Mar 31 2022 *)

def A168518(n,k,a,b,c): return (1/2)*( a*binomial(n,k) + sum( (-1)^(k-j)*(b*binomial(n+2, k-j)*(j+1)^(n+1) + 2*c*binomial(n+1,k-j)*(2*j+1)^n) for j in (0..k)) )
flatten([[A168518(n,k,-4,2,2) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 31 2022

G.f.: (1/2)*( a*(1+x)^n + b*(1-x)^(n+2)*LerchPhi(x, -n-1, 1) + c*2^(n+1)*(1 - x)^(n+1)*LerchPhi(x, -n, 1/2) ), where a = -4, b = 2, and c = 2.
From G. C. Greubel, Mar 31 2022: (Start)
T(n, k) = (1/2)*( a*binomial(n,k) + sum( (-1)^(k-j)*(b*binomial(n+2, k-j)*(j+1)^(n+1) + 2*c*binomial(n+1,k-j)*(2*j+1)^n) for j in (0..k)) ), with a = -4, b = 2, and c = 2.
T(n, n-k) = T(n, k). (End)

Edited by G. C. Greubel, Mar 31 2022

A155497 Triangle T(n, k) = binomial(n, k)binomial(n+1, k+1)binomial(2n, 2k)/(n-k+1), read by rows.

Original entry on oeis.org

1, 1, 1, 1, 18, 1, 1, 90, 90, 1, 1, 280, 1400, 280, 1, 1, 675, 10500, 10500, 675, 1, 1, 1386, 51975, 161700, 51975, 1386, 1, 1, 2548, 196196, 1471470, 1471470, 196196, 2548, 1, 1, 4320, 611520, 9417408, 22702680, 9417408, 611520, 4320, 1, 1, 6885, 1652400, 46781280, 231567336, 231567336, 46781280, 1652400, 6885, 1

Offset: 0

Roger L. Bagula, Jan 23 2009

			Triangle begins as:
  1;
  1,    1;
  1,   18,       1;
  1,   90,      90,        1;
  1,  280,    1400,      280,         1;
  1,  675,   10500,    10500,       675,         1;
  1, 1386,   51975,   161700,     51975,      1386,        1;
  1, 2548,  196196,  1471470,   1471470,    196196,     2548,       1;
  1, 4320,  611520,  9417408,  22702680,   9417408,   611520,    4320,    1;
  1, 6885, 1652400, 46781280, 231567336, 231567336, 46781280, 1652400, 6885, 1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A103371, A155495.

[Binomial(n, k)*Binomial(n+1, k+1)*Binomial(2*n, 2*k)/(n-k+1): k in [0..n], n in [0..12]]; // G. C. Greubel, May 29 2021

T[n_, k_]:= Binomial[n,k]*Binomial[n+1,k+1]*Binomial[2*n,2*k]/(n-k+1);
Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten (* modified by G. C. Greubel, May 29 2021 *)

flatten([[binomial(n, k)*binomial(n+1, k+1)*binomial(2*n, 2*k)/(n-k+1) for k in (0..12)] for n in (0..12)]) # G. C. Greubel, May 29 2021

T(n, k) = binomial(n, k)*binomial(2*n, 2*k)*f(n)/(f(k)*f(n-k)), where f(n) = (n+1)!.
T(n, k) = binomial(n, k)*binomial(n+1, k+1)*binomial(2*n, 2*k)/(n-k+1).
From G. C. Greubel, May 29 2021: (Start)
Sum_{k=0..n} T(n, k) = Hypergeometric4F3([-n, -n, -n-1, -n-1/2], [1/2, 1, 2], 1).
T(n, k) = binomial(n, k)*A155495(n, k)/(n-k+1).
T(n, k) = binomial(2*n, 2*k)*A103371(n, k)/(n-k+1). (End)

Edited by G. C. Greubel, May 29 2021

A155516 Triangle T(n, k) = binomial(2n, 2k)binomial(2n+1, 2k+1)/(2n-2*k+1), read by rows.

Original entry on oeis.org

1, 1, 1, 1, 20, 1, 1, 105, 105, 1, 1, 336, 1764, 336, 1, 1, 825, 13860, 13860, 825, 1, 1, 1716, 70785, 226512, 70785, 1716, 1, 1, 3185, 273273, 2147145, 2147145, 273273, 3185, 1, 1, 5440, 866320, 14158144, 34763300, 14158144, 866320, 5440, 1, 1, 8721, 2372112, 71954064, 367479684, 367479684, 71954064, 2372112, 8721, 1

Offset: 0

Roger L. Bagula, Jan 23 2009

			Triangle begins as:
  1;
  1,    1;
  1,   20,       1;
  1,  105,     105,        1;
  1,  336,    1764,      336,         1;
  1,  825,   13860,    13860,       825,         1;
  1, 1716,   70785,   226512,     70785,      1716,        1;
  1, 3185,  273273,  2147145,   2147145,    273273,     3185,       1;
  1, 5440,  866320, 14158144,  34763300,  14158144,   866320,    5440,    1;
  1, 8721, 2372112, 71954064, 367479684, 367479684, 71954064, 2372112, 8721, 1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A155495.

[Binomial(2*n, 2*k)*Binomial(2*n+1, 2*k+1)/(2*n-2*k+1): k in [0..n], n in [0..12]]; // G. C. Greubel, May 29 2021

T[n_, k_]:= Binomial[2*n, 2*k]*Binomial[2*n+1, 2*k+1]/(2*n-2*k+1);
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* modified by G. C. Greubel, May 29 2021 *)

flatten([[binomial(2*n, 2*k)*binomial(2*n+1, 2*k+1)/(2*n-2*k+1) for k in (0..12)] for n in (0..12)]) # G. C. Greubel, May 29 2021

T(n, k) = (2*n + 1)!!/((2*k + 1)!!*(2*(n-k) + 1)!!)*binomial(2*n, 2*k)*binomial(n, k).
From G. C. Greubel, May 29 2021: (Start)
T(n, k) = (2*n + 1)!!/((2*k + 1)!!*(2*(n-k) + 1)!!)*A155495(n, k).
T(n, k) = binomial(2*n, 2*k)*binomial(2*n+1, 2*k+1)/(2*n-2*k+1).
Sum_{k=0..n} T(n, k) = Hypergeometric4F3([-n,-n,-n-1/2,-n+1/2], [1/2,1,3/2], 1). (End)

Edited by G. C. Greubel, May 29 2021

cp's OEIS Frontend

A288470 a(n) = Sum_{k=0..n} binomial(n,k)*binomial(2*n,2*k).

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A168518 Expansion of g.f. (1/2)*( a*(1+x)^n + b*(1-x)^(n+2)*LerchPhi(x, -n-1, 1) + c*2^(n+1)*(1-x)^(n+1)*LerchPhi(x, -n, 1/2) ), where a = -4, b = 2, and c = 2, read by rows.

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A155497 Triangle T(n, k) = binomial(n, k)*binomial(n+1, k+1)*binomial(2*n, 2*k)/(n-k+1), read by rows.

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A155516 Triangle T(n, k) = binomial(2*n, 2*k)*binomial(2*n+1, 2*k+1)/(2*n-2*k+1), read by rows.

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Examples

Links

Crossrefs

Programs

Magma

Mathematica

Sage

Formula

Extensions

A288470 a(n) = Sum_{k=0..n} binomial(n,k)binomial(2n,2*k).

A168518 Expansion of g.f. (1/2)( a(1+x)^n + b(1-x)^(n+2)LerchPhi(x, -n-1, 1) + c2^(n+1)(1-x)^(n+1)*LerchPhi(x, -n, 1/2) ), where a = -4, b = 2, and c = 2, read by rows.

A155497 Triangle T(n, k) = binomial(n, k)binomial(n+1, k+1)binomial(2n, 2k)/(n-k+1), read by rows.

A155516 Triangle T(n, k) = binomial(2n, 2k)binomial(2n+1, 2k+1)/(2n-2*k+1), read by rows.