A176335 -id:A176335 - cp's OEIS frontend

A112029 a(n) = Sum_{k=0..n} binomial(n+k, k)^2.

1, 5, 46, 517, 6376, 82994, 1119210, 15475205, 217994860, 3115374880, 45035696036, 657153097330, 9663914317396, 143050882063262, 2129448324373546, 31853280798384645, 478503774600509620, 7215090439396842572, 109154411037070011504, 1656268648035559711392

Offset: 0

N. J. A. Sloane, Nov 28 2005

Vincenzo Librandi, Table of n, a(n) for n = 0..200
F. Baldassarri, S. Bosch, B. Dwork, (eds), p-adic Analysis. Lecture Notes in Mathematics, vol. 1454, pp. 194 - 204, Springer, Berlin, Heidelberg.
Matthijs J. Coster, Supercongruences.
C. Elsner, On recurrence formulas for sums involving binomial coefficients, Fib. Q., 43,1 (2005), 31-45.
Vaclav Kotesovec, Asymptotic of generalized Apery sequences with powers of binomial coefficients, Nov 04 2012
Pedro J. Miana, Hideyuki Ohtsuka, and Natalia Romero, Sums of powers of Catalan triangle numbers, arXiv:1602.04347 [math.NT], 2016.

Cf. A001700, A110197, A112028, A173774, A176335, A219562, A219563, A219564, A333592.

[(&+[Binomial(n+j, j)^2: j in [0..n]]): n in [0..20]]; // G. C. Greubel, Jul 06 2021

f := 64*x^2/(16*x-1); S := sqrt(x)*sqrt(4-x);
H := ((10*x-5/8)*hypergeom([1/4,1/4],[1],f)-(21*x-21/8)*hypergeom([1/4,5/4],[1],f))/(S*(1-16*x)^(5/4));
ord := 30;
ogf := series(int(series(H,x=0,ord),x)/S,x=0,ord);
# Mark van Hoeij, Mar 27 2013

Table[Sum[Binomial[n+k,k]^2, {k,0,n}], {n,0,20}] (* Vaclav Kotesovec, Nov 23 2012 *)

a(n) = sum(k=0, n, binomial(n+k, k)^2); \\ Michel Marcus, Jul 07 2021

[sum(binomial(n+j, j)^2 for j in (0..n)) for n in (0..20)] # G. C. Greubel, Jul 06 2021

a(n) ~ 2^(4*n+2)/(3*Pi*n). - Vaclav Kotesovec, Nov 23 2012
Recurrence: 2*(2*n+1)*(21*n-13)*n^2*a(n) = (1365*n^4 - 1517*n^3 + 240*n^2 + 216*n - 64)*a(n-1) - 4*(n-1)*(2*n-1)^2*(21*n+8)*a(n-2). - Vaclav Kotesovec, Nov 23 2012
G.f.: see Maple code. - Mark van Hoeij, Mar 27 2013
a(p-1) == 1 (mod p^3) for all primes p >= 5. See the comments in A173774. - Peter Bala, Jul 12 2024
a(n-1) = 1/(4*n) * binomial(2*n, n)^2 * ( 1 + 3*((n - 1)/(n + 1))^3 + 5*((n - 1)*(n - 2)/((n + 1)*(n + 2)))^3 + 7*((n - 1)*(n - 2)*(n - 3)/((n + 1)*(n + 2)*(n + 3)))^3 + ... )  for n >= 1. - Peter Bala, Jul 22 2024
a(m*p^r - 1) == a(m*p^(r-1) - 1) (mod p^(3*r)) for all primes p >= 5 and positive integers m and r. See Coster, Theorem 4. - Peter Bala, Nov 29 2024
a(n) = A110197(2n,n). - Alois P. Heinz, Mar 21 2025

A176331 Triangle read by rows: T(n, k) = Sum_{j=0..n} C(j, n-k) * C(j, k) * (-1)^(n - j).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 7, 7, 1, 1, 13, 28, 13, 1, 1, 21, 79, 79, 21, 1, 1, 31, 181, 315, 181, 31, 1, 1, 43, 361, 971, 971, 361, 43, 1, 1, 57, 652, 2511, 3876, 2511, 652, 57, 1, 1, 73, 1093, 5713, 12606, 12606, 5713, 1093, 73, 1, 1, 91, 1729, 11789, 35246, 50358, 35246, 11789, 1729, 91, 1

Offset: 0

Paul Barry, Apr 15 2010

			Triangle begins
  1;
  1,  1;
  1,  3,    1;
  1,  7,    7,     1;
  1, 13,   28,    13,     1;
  1, 21,   79,    79,    21,     1;
  1, 31,  181,   315,   181,    31,     1;
  1, 43,  361,   971,   971,   361,    43,     1;
  1, 57,  652,  2511,  3876,  2511,   652,    57,    1;
  1, 73, 1093,  5713, 12606, 12606,  5713,  1093,   73,  1;
  1, 91, 1729, 11789, 35246, 50358, 35246, 11789, 1729, 91, 1;

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Row sums are A176332.
Diagonal sums are A176334.
Central coefficients T(2*n, n) are A176335.

T:= function(n,k)
    return Sum([0..n], j-> (-1)^(n-j)*Binomial(j,k)*Binomial(j,n-k) );
  end;
Flat(List([0..10], n-> List([0..n], k-> T(n,k) ))); # G. C. Greubel, Dec 07 2019

T:= func< n,k | &+[(-1)^(n-j)*Binomial(j,n-k)*Binomial(j,k): j in [0..n]] >;
[T(n,k): k in [0..n], n in [0..10]]; // G. C. Greubel, Dec 07 2019

T:= proc(n, k) option remember; add( (-1)^(n-j)*binomial(j, n-k)*binomial(j, k), j=0..n); end: seq(seq(T(n, k), k=0..n), n=0..10); # G. C. Greubel, Dec 07 2019
T := (n, k) -> binomial(n, k)^2*hypergeom([1, -k, -n + k], [-n, -n], -1):
seq(seq(simplify(T(n, k)), k = 0..n), n = 0..9); # Peter Luschny, May 13 2024

T[n_, k_]:= Sum[(-1)^(n-j)*Binomial[j, k]*Binomial[j, n-k], {j,0,n}]; Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Dec 07 2019 *)

T(n,k) = sum(j=0, n, (-1)^(n-j)*binomial(j, n-k)*binomial(j, k)); \\ G. C. Greubel, Dec 07 2019

@CachedFunction
def T(n, k): return sum( (-1)^(n-j)*binomial(j, n-k)*binomial(j, k) for j in (0..n))
[[T(n, k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Dec 07 2019

T(n, k) = T(n, n-k).

T(n, k) = binomial(n, k)^2*hypergeom([1, -k, -n + k], [-n, -n], -1). - Peter Luschny, May 13 2024

A176334 Diagonal sums of number triangle A176331.

Original entry on oeis.org

1, 1, 2, 4, 9, 21, 51, 124, 305, 755, 1879, 4698, 11792, 29694, 74984, 189811, 481498, 1223713, 3115200, 7942134, 20275362, 51823246, 132604193, 339644739, 870745187, 2234208932, 5737129623, 14742751524, 37909928908, 97543380598

Offset: 0

Paul Barry, Apr 15 2010

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A176331, A176332, A176335.

T:= function(n,k)
    return Sum([0..n], j-> (-1)^(n-j)*Binomial(j,k)*Binomial(j,n-k) );
  end;
List([0..30], n-> Sum([0..Int(n/2)], j-> T(n-j,j) )); # G. C. Greubel, Dec 07 2019

T:= func< n,k | &+[(-1)^(n-j)*Binomial(j,n-k)*Binomial(j,k): j in [0..n]] >;
[(&+[T(n-k,k): k in [0..Floor(n/2)]]): n in [0..30]]; // G. C. Greubel, Dec 07 2019

A176334 := proc(n)
    add(add(binomial(j,n-2*k)*binomial(j,k)*(-1)^(n-k-j),j=0..n-k), k=0..floor(n/2)) ;
end proc: # R. J. Mathar, Feb 10 2015

T[n_, k_]:= T[n, k]= Sum[(-1)^(n-j)*Binomial[j, k]*Binomial[j, n-k], {j,0,n}]; Table[Sum[T[n-k, k], {k,0,Floor[n/2]}], {n,0,30}] (* G. C. Greubel, Dec 07 2019 *)

T(n,k) = sum(j=0, n, (-1)^(n-j)*binomial(j, n-k)*binomial(j, k));
vector(30, n, sum(j=0, (n-1)\2, T(n-j-1,j)) ) \\ G. C. Greubel, Dec 07 2019

@CachedFunction
def T(n, k): return sum( (-1)^(n-j)*binomial(j, n-k)*binomial(j, k) for j in (0..n))
[sum(T(n-k, k) for k in (0..floor(n/2))) for n in (0..30)] # G. C. Greubel, Dec 07 2019

a(n) = Sum_{k=0..floor(n/2)} Sum_{j=0..n-k} C(j,n-2k)*C(j,k)*(-1)^(n-k-j).

a(n) ~ phi^(2*n+3) / (4*5^(1/4)*sqrt(Pi*n)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Aug 08 2024

A375178 a(n) = Sum_{k = 0..n-1} binomial(n+k-1, k)^3 (same as A112028 with an extra 0 at the start).

Original entry on oeis.org

0, 1, 9, 244, 9065, 389376, 18188478, 897376152, 46011772521, 2427553965160, 130930630643384, 7186614533569296, 400132290102421214, 22543708920891189136, 1282873288801683197250, 73628947696550668509744, 4257138240245923453355625, 247733479854085081062353400, 14498252738780732999484606360

Offset: 0

Peter Bala, Aug 03 2024

Compare with the identity Sum_{k = 0..n-1} binomial(n+k-1, k) = (1/2) * binomial(2*n, n) = (1/2) * A000984(n) for n >= 1.
The central binomial coefficients satisfy the supercongruence (1/2) * binomial(2*p, p) == 1 (mod p^3) for all primes p >= 5 (Wolstenholme's theorem).
For prime p, binomial(p+k-1, k) == 0 (mod p) for 1 <= k <= p-1. It follows that a(p) == 1 (mod p^3) for all primes p. We conjecture that, in fact, the stronger congruence a(p) == 1 (mod p^5) holds for all primes p >= 7.
Further, we conjecture that for r >= 2 and prime p >= 5, a(p^r) == a(p^(r-1)) (mod p^(3*r+3)).
More generally, for a positive integer m, define a sequence {b_m(n) : n >= 0} by setting b_m(n) = Sum_{k = 0..n-1} binomial(n+k-1, k)^(2*m+1). Then the congruence b_m(p) == 1 (mod p^(2*m+1)) clearly holds for all primes p. We conjecture that the stronger supercongruence b_m(p) == 1 (mod p^(2*m+3)) holds for all primes p >= 2*m + 5, and for r >= 2, the supercongruence b_m(p^r) == b_m(p^(r-1)) (mod p^(3*r+2*m+1)) also holds for all primes p >= 2*m + 5.
Essentially a duplicate of A112028.

			Examples of supercongruences:
a(7) - a(1) = 897376152 - 1 = (7^5)*107*499 == 0 (mod 7^5)
a(11) - a(1) = 7186614533569296 - 1 = 5*(11^5)*8924644409 == 0 (mod 11^5).

Romeo Meštrović, Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2012), arXiv:1111.3057 [math.NT], (2011).

Cf. A000984, A010763, A112028, A176335, A375179, A375180.

seq(add( binomial(n+k-1, k)^3, k = 0..n-1), n = 0..20);

a(n) = Sum_{k = 0..n-1} (-1)^k * binomial(-n, k)^3.

a(n) ~ 2^(6*n-3)/(7*Pi^(3/2)*n^(3/2)). - Vaclav Kotesovec, Aug 03 2024

A382849 a(n) = Sum_{k=0..n} (-1)^(n-k) * (binomial(n,k) * binomial(n+k,k))^2.

Original entry on oeis.org

1, 3, 1, -357, -6999, -62997, 444529, 27783003, 508019689, 3206511003, -89889084999, -3274278527517, -49395223500999, -66079827133317, 16197028704290001, 433384098559415643, 4988878584849669609, -35687369703800052357, -2815548294132454060151, -58942279760573467233357

Offset: 0

Ilya Gutkovskiy, Apr 06 2025

sign

Diagonal of the rational function 1 / (1 - y + z + x*y + z*w + x*z + x*y*w + x*y*z*w).

Cf. A005258, A005259, A126869, A176335, A228304, A382848.

Table[Sum[(-1)^(n - k) (Binomial[n, k] Binomial[n + k, k])^2, {k, 0, n}], {n, 0, 19}]
Table[(-1)^n HypergeometricPFQ[{-n, -n, n + 1, n + 1}, {1, 1, 1}, -1], {n, 0, 19}]
Table[SeriesCoefficient[1/(1 - y + z + x y + z w + x z + x y w + x y z w), {x, 0, n}, {y, 0, n}, {z, 0, n}, {w, 0, n}], {n, 0, 19}]

cp's OEIS Frontend

A112029 a(n) = Sum_{k=0..n} binomial(n+k, k)^2.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Sage

Formula

A176331 Triangle read by rows: T(n, k) = Sum_{j=0..n} C(j, n-k) * C(j, k) * (-1)^(n - j).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

A176334 Diagonal sums of number triangle A176331.

Views

Author

Keywords

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

A375178 a(n) = Sum_{k = 0..n-1} binomial(n+k-1, k)^3 (same as A112028 with an extra 0 at the start).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Formula

A382849 a(n) = Sum_{k=0..n} (-1)^(n-k) * (binomial(n,k) * binomial(n+k,k))^2.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica