A208425 -id:A208425 - cp's OEIS frontend

A275027 a(n) = Sum_{k=0..n} C(n,k)^2C(n-k,k), where C(n,k) denotes the binomial coefficient n!/(k!(n-k)!).

1, 1, 5, 19, 85, 401, 1931, 9605, 48469, 248365, 1286605, 6726875, 35441275, 187935775, 1002122525, 5369287019, 28889315669, 156015203845, 845330354321, 4593724615175, 25029614166685, 136704935601785, 748273234994675, 4103928115592365, 22549175326327675, 124105065258631651, 684100888645922051, 3776354280849020005

Offset: 0

Zhi-Wei Sun, Nov 12 2016

nonn

Conjecture: For any prime p > 5 and positive integer n, the number (a(p*n)-a(n))/(p*n)^3 is always a p-adic integer.
The author has proved that for any prime p > 5 and positive integer n the number (a(p*n)-a(n))/(p^3*n^2) is always a p-adic integer.
As a(n) = Sum_{k=0..n} C(n,k)*C(n,2k)*C(2k,k) and C(2k,k) = 2*C(2k-1,k-1) for k = 1,2,3,..., we see that a(n) is always odd. We guess that a(n) is congruent to one of 0, 1, -1 modulo 5.
Diagonal of the rational function 1 / ((1 - x)*(1 - y)*(1 - z) - x^2*y*z). - Ilya Gutkovskiy, Apr 23 2025

			a(2) = 5 since a(2) = Sum_{k=0,1,2}C(2,k)^2*C(2-k,k) = C(2,0)^2*C(2,0) + C(2,1)^2*C(1,1) = 1 + 4 = 5.

Zhi-Wei Sun, Table of n, a(n) for n = 0..200
Joel A. Henningsen and Armin Straub, Generalized Lucas congruences and linear p-schemes, arXiv:2111.08641 [math.NT], 2021.
Zhi-Wei Sun, Supercongruences involving Lucas sequences, arXiv:1610.03384 [math.NT], 2016.

Cf. A000984, A005258, A208425, A244973, A277640.

a[n_]:=a[n]=Sum[Binomial[n,k]^2*Binomial[n-k,k],{k,0,n/2}]
Table[a[n],{n,0,27}]
a[n_] := HypergeometricPFQ[{-n, 1/2 - n/2, -n/2}, {1, 1}, -4];
Table[a[n], {n, 0, 27}] (* Peter Luschny, Mar 21 2018 *)

a(n) = sum(k=0, n, binomial(n,k)^2*binomial(n-k,k)); \\ Michel Marcus, Nov 13 2016

a(n) = Sum_{k=0..n}C(n,k)*C(n,2k)*C(2k,k).
By the Zeilberger algorithm, we have the recurrence (n+3)^2*(23n+25)*a(n+3) = 25*(n+1)^2*(23n+48)*a(n) + (391n^3+1989n^2+3288n+1750)*a(n+1) + (46n^3+280n^2+ 519n+265)*a(n+2) for all n >= 0.
a(n) = hypergeom([-n, 1/2 - n/2, -n/2], [1, 1], -4). - Peter Luschny, Mar 21 2018
a(n) ~ c * d^n / (Pi*n), where d = 5.729031537980930837932235459792820714... is the real root of the equation -25 - 17*d - 2*d^2 + d^3 = 0 and c = 1.107089291883984657933126801836156175486638498732... is the positive real root of the equation -125 + 1048*c^2 - 2576*c^4 + 1472*c^6 = 0. - Vaclav Kotesovec, Jun 09 2019
G.f.: hypergeom([1/12, 5/12],[1],-1728*(25*x^3+17*x^2+2*x-1)*x^7/(1-4*x-10*x^2+4*x^3+25*x^4)^3)/(1-4*x-10*x^2+4*x^3+25*x^4)^(1/4). - Mark van Hoeij, Nov 28 2024

A274783 Diagonal of the rational function 1/(1 - (wxy*z + wxy + wxz + wyz + xyz)).

Original entry on oeis.org

1, 1, 1, 25, 121, 361, 3361, 24361, 116425, 790441, 6060121, 36888721, 238815721, 1760983225, 11968188961, 79763351305, 570661612585, 4040282139625, 27901708614985, 198090585115105, 1420583920034161, 10056659775872161, 71730482491962361, 517012699162717825, 3713833648541268121

Offset: 0

Gheorghe Coserea, Jul 13 2016

Diagonal of the rational function 1/(1 - (x^3 + y^3 + z^3 + w^3 + x*y*z*w)). - Seiichi Manyama, Jul 04 2025

Seiichi Manyama, Table of n, a(n) for n = 0..1000
A. Bostan, S. Boukraa, J.-M. Maillard, J.-A. Weil, Diagonals of rational functions and selected differential Galois groups, arXiv preprint arXiv:1507.03227 [math-ph], 2015.
S. Eger, On the Number of Many-to-Many Alignments of N Sequences, arXiv:1511.00622 [math.CO], 2015.
Jacques-Arthur Weil, Supplementary Material for the Paper "Diagonals of rational functions and selected differential Galois groups"

Cf. A268545-A268555.
Cf. A001850, A208425, A361636.
Cf. A082488, A274785, A361637.

with(combinat):
seq(add((n+k)!/(k!^4*(n-3*k)!), k = 0..floor(n/3)), n = 0..20); # Peter Bala, Jan 27 2018

my(x='x, y='y, z='z, w='w);
R = 1/(1-(w*x*y*z+w*x*y+w*x*z+w*y*z+x*y*z));
diag(n, expr, var) = {
  my(a = vector(n));
  for (i = 1, #var, expr = taylor(expr, var[#var - i + 1], n));
  for (k = 1, n, a[k] = expr;
       for (i = 1, #var, a[k] = polcoeff(a[k], k-1)));
  return(a);
};
diag(20, R, [x,y,z,w])

0 = x^2*(x+3)^2*(x^4 - 260*x^3 + 6*x^2 - 4*x + 1)*y''' + 3*x*(x+3)*(2*x^5 - 381*x^4 - 1944*x^3 + 34*x^2 - 18*x + 3)*y'' + (7*x^6 - 764*x^5 - 9101*x^4 - 27264*x^3 + 381*x^2 - 132*x + 9)*y' + (x^5 - 13*x^4 - 246*x^3 - 5946*x^2 + 69*x - 9)*y, where y is the g.f.
a(n) = Sum_{k = 0..floor(n/3)} (n+k)!/(k!^4*(n-3*k)!) = Sum_{k = 0..floor(n/3)} binomial(n,3*k)*binomial(n+k,k)*(3*k)!/k!^3 (apply Eger, Theorem 3 to the set of column vectors S = {[1,1,1,1], [1,1,0,1], [1,0,1,1], [0,1,1,1], [1,1,1,0]}). - Peter Bala, Jan 27 2018
G.f.: Sum_{k>=0} (4*k)!/k!^4 * x^(3*k)/(1-x)^(4*k+1). - Seiichi Manyama, Mar 19 2023
From Vaclav Kotesovec, Mar 19 2023: (Start)
Recurrence: n^3*(2*n - 5)*(4*n - 11)*(4*n - 7)*a(n) = (4*n - 11)*(32*n^5 - 184*n^4 + 368*n^3 - 327*n^2 + 147*n - 27)*a(n-1) - (192*n^6 - 1920*n^5 + 7628*n^4 - 15366*n^3 + 16567*n^2 - 9117*n + 2025)*a(n-2) + (4*n - 9)*(4*n - 3)*(520*n^4 - 4420*n^3 + 13809*n^2 - 18769*n + 9367)*a(n-3) - (n-3)^3*(2*n - 3)*(4*n - 7)*(4*n - 3)*a(n-4).
a(n) ~ sqrt(9/8 + 3/(32*sqrt(2)) + sqrt(1085/32 + 161/(2*sqrt(2)))/8) * (1 + 2*sqrt(2) + 2*sqrt(2*(2*sqrt(2) - 1)))^n / (Pi^(3/2) * n^(3/2)). (End)

A361636 Diagonal of the rational function 1/(1 - vwx*yz (1 + 1/v + 1/w + 1/x + 1/y + 1/z)).

Original entry on oeis.org

1, 1, 1, 1, 121, 721, 2521, 6721, 128521, 1277641, 7539841, 32527441, 281835841, 3031468441, 23779315561, 139431015361, 962322302761, 9034098300361, 79726215362761, 569831799431881, 3952559737085401, 32660742079719601, 289694072383115401

Offset: 0

Seiichi Manyama, Mar 19 2023

nonn

Diagonal of the rational function 1/(1 - (v^4 + w^4 + x^4 + y^4 + z^4 + v*w*x*y*z)). - Seiichi Manyama, Jul 04 2025

Winston de Greef, Table of n, a(n) for n = 0..1064

Cf. A001850, A208425, A274783.
Cf. A008978.
Cf. A082489, A361703.

Table[Sum[(n + k)!/(k!^5*(n - 4*k)!), {k, 0, n/4}], {n, 0, 25}] (* Vaclav Kotesovec, Mar 19 2023 *)

a(n) = sum(k=0, n\4, (n+k)!/(k!^5*(n-4*k)!));

a(n) = Sum_{k=0..floor(n/4)} (n+k)!/(k!^5 * (n-4*k)!).
G.f.: Sum_{k>=0} (5*k)!/k!^5 * x^(4*k)/(1-x)^(5*k+1).
Recurrence: n^4*(5*n - 19)*(5*n - 18)*(5*n - 17)*(5*n - 14)*(5*n - 13)*(5*n - 9)*a(n) = (5*n - 19)*(5*n - 18)*(5*n - 14)*(625*n^7 - 6125*n^6 + 23025*n^5 - 43195*n^4 + 45394*n^3 - 28716*n^2 + 10144*n - 1536)*a(n-1) - (5*n - 19)*(31250*n^9 - 568750*n^8 + 4441875*n^7 - 19516000*n^6 + 53172025*n^5 - 93366740*n^4 + 106140132*n^3 - 75781664*n^2 + 30987264*n - 5529600)*a(n-2) + (5*n - 4)*(31250*n^9 - 725000*n^8 + 7354375*n^7 - 42784750*n^6 + 157237100*n^5 - 378480620*n^4 + 596812963*n^3 - 594970390*n^2 + 340845072*n - 85743360)*a(n-3) + (5*n - 16)*(5*n - 9)*(5*n - 8)*(5*n - 4)*(78000*n^6 - 1450800*n^5 + 11179085*n^4 - 45672814*n^3 + 104341702*n^2 - 126378083*n + 63400710)*a(n-4) + (n-4)^4*(5*n - 14)*(5*n - 13)*(5*n - 12)*(5*n - 9)*(5*n - 8)*(5*n - 4)*a(n-5). - Vaclav Kotesovec, Mar 19 2023

A208426 Expansion of Sum_{n>=0} (3n)!/n!^3 x^(2n)/(1-3x)^(3*n+1).

Original entry on oeis.org

1, 3, 15, 99, 711, 5373, 42099, 338355, 2771127, 23028813, 193610385, 1643215005, 14056350075, 121040308665, 1048212778635, 9122168556819, 79727173530327, 699443806767525, 6156776010386481, 54356715121718349, 481194980656865721, 4270165015550478003

Offset: 0

Paul D. Hanna, Feb 26 2012

nonn

Compare g.f. to: Sum_{n>=0} (3*n)!/n!^3 * x^(2*n)/(1-2*x)^(3*n+1), which is a g.f. of the Franel numbers (A000172).
Diagonal of rational functions 1/(1 - x*y - y*z - x*z - 3*x*y*z), 1/(1 - x*y + y*z + x*z - 3*x*y*z). - Gheorghe Coserea, Jul 04 2018
Diagonal of the rational function 1/(1 - (x^2 + y^2 + z^2 + 3*x*y*z)). - Seiichi Manyama, Jul 05 2025

			G.f.: A(x) = 1 + 3*x + 15*x^2 + 99*x^3 + 711*x^4 + 5373*x^5 + 42099*x^6 + ...
where
A(x) = 1/(1-3*x) + 6*x^2/(1-3*x)^4 + 90*x^4/(1-3*x)^7 + 1680*x^6/(1-3*x)^10 + 34650*x^8/(1-3*x)^13 + 756756*x^10/(1-3*x)^16 + ...

Gheorghe Coserea, Table of n, a(n) for n = 0..200

Cf. A000172, A002893, A069835, A208425, A294035.

Table[3^n * HypergeometricPFQ[{1/2 - n/2, -n/2, 1 + n}, {1, 1}, 4/9], {n, 0, 25}] (* Vaclav Kotesovec, Oct 07 2020 *)

{a(n)=polcoeff(sum(m=0,n, (3*m)!/m!^3*x^(2*m)/(1-3*x+x*O(x^n))^(3*m+1)),n)}
for(n=0,31,print1(a(n),", "))

a(n) = sum(k=0, n\2, (n+k)!/(k!^3*(n-2*k)!) * 3^(n-2*k)); \\ Gheorghe Coserea, Jul 04 2018

From Gheorghe Coserea, Jul 04 2018: (Start)
a(n) = Sum_{k=0..floor(n/2)} (n+k)!/(k!^3*(n-2*k)!) * 3^(n-2*k).
G.f. y=A(x) satisfies: 0 = x*(3*x + 2)*(27*x^3 + 9*x - 1)*y'' + (243*x^4 + 216*x^3 + 27*x^2 + 36*x - 2)*y' + 3*(27*x^3 + 33*x^2 - 2*x + 2)*y.
(End)
From Vaclav Kotesovec, Oct 07 2020: (Start)
Recurrence: n^2*(3*n - 5)*a(n) = 3*(9*n^3 - 24*n^2 + 17*n - 4)*a(n-1) + 3*(3*n - 4)*a(n-2) + 27*(n-2)^2*(3*n - 2)*a(n-3).
a(n) ~ sqrt(2 + sqrt(5)*phi^(-1/3) + sqrt(5)*phi^(1/3)) * 3^n * (1 + phi^(-2/3) + phi^(2/3))^n / (2*Pi*n), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio.
(End)

A278405 a(n) = Sum_{k=0..n} binomial(n,2k)^2*binomial(n-k,k).

Original entry on oeis.org

1, 1, 2, 19, 110, 476, 2477, 15093, 86830, 485290, 2826902, 16857116, 100034453, 594833357, 3574477090, 21611465819, 130955824174, 796195223398, 4860425688176, 29760574848750, 182655048136510, 1123720751229858, 6929124085148938, 42811398244528788

Offset: 0

Zhi-Wei Sun, Nov 20 2016

nonn

Conjecture: For any prime p > 5 and positive integer n, the number (a(p*n)-a(n))/(p*n)^3 is always a p-adic integer.
We have proved that for any prime p > 5 and positive integer n the number (a(p*n)-a(n))/(p^3*n^2) is always a p-adic integer.
Diagonal of the rational function 1 / ((1 + x)*(1 - x)*(1 - y)*(1 - z) - x*y*z). - Ilya Gutkovskiy, Apr 23 2025

			a(3) = 19 since a(3) = C(3,2*0)^2*C(3-0,0) + C(3,2*1)^2*C(3-1,1) = 1 + 3^2*2 = 19.
G.f. = 1 + x + 2*x^2 + 19*x^3 + 110*x^4 + 476*x^5 + 2477*x^6 + 15093*x^7 + ...

Zhi-Wei Sun, Table of n, a(n) for n = 0..200
Zhi-Wei Sun, Supercongruences involving Lucas sequences, arXiv:1610.03384 [math.NT], 2016.

Cf. A208425, A244973, A275027, A277640, A278415.

a[n_]:=a[n]=Sum[Binomial[n,2k]^2*Binomial[n-k,k],{k,0,n/2}]
Table[a[n],{n,0,27}]

A278415 a(n) = Sum_{k=0..n} binomial(n, 2k)binomial(n-k, k)(-1)^k.

Original entry on oeis.org

1, 1, 0, -5, -16, -24, 15, 197, 576, 724, -1200, -8832, -22801, -21293, 76440, 408795, 922368, 499104, -4446588, -19025060, -37012416, -1673992, 245604832, 880263936, 1441226991, -908700649, -13088509200, -40222012703, -52991533744, 88167061704, 678172355415, 1805175708261, 1747974632448, -6237554623536, -34300087628480

Offset: 0

Zhi-Wei Sun, Nov 21 2016

sign

Conjecture: For any prime p > 3 and positive integer n, the number (a(p*n)-a(n))/(p*n)^2 is always a p-adic integer.
We are able to show that for any prime p > 3 and positive integer n the number (a(p*n)-a(n))/(p^2*n) is always a p-adic integer.
See also A275027 and A278405 for similar conjectures.

			a(3) = -5 since a(3) = C(3, 2*0)*C(3-0, 0)(-1)^0 + C(3,2*1)*C(3-1,1)(-1)^1 = 1 - 6 = -5.

Zhi-Wei Sun, Table of n, a(n) for n = 0..400
Zhi-Wei Sun, Supercongruences involving Lucas sequences, arXiv:1610.03384 [math.NT], 2016.

Cf. A208425, A244973, A277640, A275027, A278405.

a[n_]:=Sum[Binomial[n,2k]Binomial[n-k,k](-1)^k,{k,0,n}]
Table[a[n],{n,0,34}]

cp's OEIS Frontend

A275027 a(n) = Sum_{k=0..n} C(n,k)^2*C(n-k,k), where C(n,k) denotes the binomial coefficient n!/(k!*(n-k)!).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A274783 Diagonal of the rational function 1/(1 - (w*x*y*z + w*x*y + w*x*z + w*y*z + x*y*z)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

PARI

Formula

A361636 Diagonal of the rational function 1/(1 - v*w*x*y*z * (1 + 1/v + 1/w + 1/x + 1/y + 1/z)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A208426 Expansion of Sum_{n>=0} (3*n)!/n!^3 * x^(2*n)/(1-3*x)^(3*n+1).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A278405 a(n) = Sum_{k=0..n} binomial(n,2k)^2*binomial(n-k,k).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A278415 a(n) = Sum_{k=0..n} binomial(n, 2k)*binomial(n-k, k)*(-1)^k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A275027 a(n) = Sum_{k=0..n} C(n,k)^2C(n-k,k), where C(n,k) denotes the binomial coefficient n!/(k!(n-k)!).

A274783 Diagonal of the rational function 1/(1 - (wxy*z + wxy + wxz + wyz + xyz)).

A361636 Diagonal of the rational function 1/(1 - vwx*yz (1 + 1/v + 1/w + 1/x + 1/y + 1/z)).

A208426 Expansion of Sum_{n>=0} (3n)!/n!^3 x^(2n)/(1-3x)^(3*n+1).

A278415 a(n) = Sum_{k=0..n} binomial(n, 2k)binomial(n-k, k)(-1)^k.