A112028 -id:A112028 - cp's OEIS frontend

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

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

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

Original entry on oeis.org

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

A219562 a(n) = Sum_{k=0..n} binomial(n+k,k)^4.

Original entry on oeis.org

1, 17, 1378, 170257, 25561876, 4294835666, 776487013506, 147812510671121, 29234435383857304, 5955068493838815892, 1241820686691538181636, 263946916625793118532050, 56996643356459050103185444, 12473214064899644269110156626, 2760963661677614009262282769378

Offset: 0

Vaclav Kotesovec, Nov 23 2012

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 Coster, Supercongruences.
Vaclav Kotesovec, Asymptotic of generalized Apery sequences with powers of binomial coefficients, Nov 04 2012

Cf. A001700, A112029, A112028, A219563, A219564.

q := x-4+I*((x+4)*(16-x))^(1/2);
f := x*(q/8)^4;
s := ((q-2)/(8*I-6))^(1/4);
y1 := hypergeom([1/8, 1/8], [3/4], f) * s / x^(1/8);
r := 2/(x*((x+4)*(16-x))^(1/2)*y1^2);
h := hypergeom([1/2, 1/2, 1/2, 1/2],[1, 1, 1],256*x);
u := (15*(223*x+72)*x^2*diff(h,x,x,x)+(14579*x+3226)*x*diff(h,x,x)
+(9969*x+1002)*diff(h,x)+320*h)/(16*(16-x)*(x+4)*x^2);
ogf := y1^2*Int(r*(1+Int(r*Int(u/(r*y1)^2,x),x)),x) ;
# Check o.g.f. by computing a series expansion:
SER := proc(a,x) series(a,x,20) end:
INT := proc(a,x) int(SER(a,x),x) end:
SER(eval(ogf, Int = INT),x); # Mark van Hoeij, Apr 02 2013

Table[Sum[Binomial[n+k,k]^4, {k,0,n}], {n,0,20}]

a(n) = sum(k=0, n, binomial(n+k,k)^4); \\ Michel Marcus, Jul 15 2022

a(n) ~ 2^(8*n+4)/(15*Pi^2*n^2).
Recurrence: 4*(n-1)*(4*n-1)*(4*n+1)*(279825*n^6 - 2240985*n^5 + 7416081*n^4 - 12962383*n^3 + 12597634*n^2 - 6438500*n + 1347304)*n^4*a(n) = 2*(n-1)*(2290647450*n^12 - 22926837585*n^11 + 100717526436*n^10 - 254986993727*n^9 + 410380920831*n^8 - 435959897978*n^7 + 305660392723*n^6 - 134977315842*n^5 + 31413259700*n^4 + 2833672*n^3 - 2076143616*n^2 + 500898816*n - 39813120)*a(n-1) + (859902225*n^13 - 10755967005*n^12 + 60090860763*n^11 - 197381561581*n^10 + 422055067481*n^9 - 613861172995*n^8 + 615013106513*n^7 - 418396400175*n^6 + 182810864162*n^5 - 42759392772*n^4 + 146171272*n^3 + 2813432832*n^2 - 691172352*n + 55738368)*a(n-2) - 16*(n-2)^3*(2*n-3)^4*(279825*n^6 - 562035*n^5 + 408531*n^4 - 111409*n^3 - 5504*n^2 + 7968*n - 1024)*a(n-3).
G.f. as an expression in terms of 2F1 and 4F3 functions is given in the Maple program below. - Mark van Hoeij, Apr 02 2013
From Peter Bala, Nov 29 2024: (Start)
Conjecture: a(p-1) == 1 (mod p^5) for prime p >= 7 (checked up to p = 499). Coster, Theorem 4, proves that a(p-1) == 1 (mod p^3) for primes p >= 5.
Conjecture: for r >= 2, the supercongruence a(p^r - 1) == a(p^(r-1) - 1) (mod p^(3*r+3)) may hold for all primes p >= 5. Coster, Theorem 4, proves that a(p^r -1) == a(p^(r-1) - 1) (mod p^(3*r)) for r >= 2 and all primes p >= 5. (End)

A219563 Sum(binomial(n+k,k)^5, k=0..n).

Original entry on oeis.org

1, 33, 8020, 3301025, 1733984376, 1048567813062, 694995078406056, 491336887915201185, 364377975224032162000, 280380150421755638519408, 222165159124597435189467696, 180288439972217748901049985158, 149230751849318301857448761484400, 125602423480863080624602495191566250

Offset: 0

Vaclav Kotesovec, Nov 23 2012

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..200
V. Kotesovec, Asymptotic of generalized Apery sequences with powers of binomial coefficients, Nov 04 2012

Cf. A001700, A112029, A112028, A219562, A219564.

Table[Sum[Binomial[n+k,k]^5, {k,0,n}], {n,0,20}]

a(n) ~ 2^(10*n+5)/(31*(Pi*n)^(5/2)).

A219564 Sum(binomial(n+k,k)^6, k=0..n).

Original entry on oeis.org

1, 65, 47386, 65004097, 119498671876, 260128695981674, 632156164654144530, 1659900189891175027265, 4616088190888638302435080, 13418259230056806455830305940, 40401802613222456104862752944356, 125182282922559710456869140648653290, 397195659937314116991934285462527257236

Offset: 0

Vaclav Kotesovec, Nov 23 2012

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..200
V. Kotesovec, Asymptotic of generalized Apery sequences with powers of binomial coefficients, Nov 04 2012

Cf. A001700 (q=1), A112029 (q=2), A112028 (q=3), A219562 (q=4), A219563 (q=5).

Table[Sum[Binomial[n+k,k]^6, {k,0,n}], {n,0,20}]

a(n) ~ 2^(12*n+6)/(63*Pi^3*n^3)

Generally (for q > 0), Sum_{k=0..n} C(n + k,k)^q is asymptotic to 2^((2*n+1)*q)/((2^q-1)*(Pi*n)^(q/2)) * (1 - q/(2*n)*(1/4+1/(2^q-1)^2) + O(1/n^2))

A295612 a(n) = Sum_{k=0..n} binomial(n+k,k)^k.

Original entry on oeis.org

1, 3, 40, 8105, 24053106, 1016507243472, 622366942086680904, 5608321882919220905812521, 752711651805019773658037206391596, 1518219710649896586598445898967340890577318, 46343146356260529633020448755386347142785083052620084

Offset: 0

Ilya Gutkovskiy, Nov 24 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..41
Eric Weisstein's World of Mathematics, Binomial Sums

Cf. A001700, A046899, A112028, A112029, A167008, A219562, A219563, A219564.

Table[Sum[Binomial[n + k, k]^k, {k, 0, n}], {n, 0, 10}]
Table[Sum[((n + k)!/(n! k!))^k, {k, 0, n}], {n, 0, 10}]

a(n) = sum(k=0, n, binomial(n+k,k)^k); \\ Michel Marcus, Nov 25 2017

a(n) = Sum_{k=0..n} A046899(n,k)^k.

a(n) ~ 2^(2*n^2) / (exp(1/8) * Pi^(n/2) * n^(n/2)). - Vaclav Kotesovec, Nov 25 2017

A336829 a(n) = Sum_{k=0..n} binomial(n+k,k)^n.

Original entry on oeis.org

1, 3, 46, 9065, 25561876, 1048567813062, 632156164654144530, 5652307059542612442465921, 755658094192422806457805924637704, 1521188219372604726826961340683399629967888, 46388428590466766659538640978460161019178279424832676

Offset: 0

Ilya Gutkovskiy, Aug 05 2020

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..41

Cf. A001700, A112028, A112029, A167010, A219562, A219563, A219564.

[(&+[Binomial(2*n-j,n)^n: j in [0..n]]): n in [0..20]]; // G. C. Greubel, Aug 31 2022

Table[Sum[Binomial[n + k, k]^n, {k, 0, n}], {n, 0, 10}]

a(n) = sum(k=0, n, binomial(n+k, k)^n); \\ Michel Marcus, Aug 05 2020

def A336829(n): return sum(binomial(2*n-j, n)^n for j in (0..n))
[A336829(n) for n in (0..20)] # G. C. Greubel, Aug 31 2022

a(n) ~ exp(-1/8) * 2^(2*n^2) / (Pi*n)^(n/2). - Vaclav Kotesovec, Jul 10 2021

A374676 a(n) = Sum_{k=0..n} binomial(n+k-1,n) * binomial(n+k,n)^2.

Original entry on oeis.org

1, 4, 117, 4416, 191025, 8959098, 443175257, 22764880288, 1202693106969, 64935435950760, 3567189106107044, 198746486074429164, 11203798260525398593, 637866038409406067394, 36624374381748740836905, 2118319467225018572438976, 123307986154526506959597225

Offset: 0

Seiichi Manyama, Jul 16 2024

nonn

Cf. A001791, A374675.

Cf. A112028.

a(n) = sum(k=0, n, binomial(n+k-1, n)*binomial(n+k, n)^2);

a(n) = Sum_{k=0..n} (k/(n+k)) * binomial(n+k,k)^3 for n > 0.

cp's OEIS Frontend

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

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

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A219562 a(n) = Sum_{k=0..n} binomial(n+k,k)^4.

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A219563 Sum(binomial(n+k,k)^5, k=0..n).

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A219564 Sum(binomial(n+k,k)^6, k=0..n).

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A295612 a(n) = Sum_{k=0..n} binomial(n+k,k)^k.

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Mathematica

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A336829 a(n) = Sum_{k=0..n} binomial(n+k,k)^n.

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A374676 a(n) = Sum_{k=0..n} binomial(n+k-1,n) * binomial(n+k,n)^2.

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