A346646 -id:A346646 - cp's OEIS frontend

A378327 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(nk,k) / ((n-1)k + 1).

1, 2, 5, 25, 257, 4361, 104425, 3241316, 123865313, 5628753361, 296671566941, 17798975341467, 1197924420178381, 89394126594968755, 7326377073291002147, 654215578855903951141, 63225054646397348577601, 6575059243843086616460321, 732138834180570978286488133

Offset: 0

Vaclav Kotesovec, Nov 23 2024

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..338

Cf. A007317, A007556, A188687, A346646, A346647, A346648, A346649, A346650.

Cf. A378326.

Table[Sum[Binomial[n, k] Binomial[n*k, k]/((n-1)*k + 1), {k, 0, n}], {n, 0, 20}]

a(n) ~ exp(n + exp(-1) - 1/2) * n^(n - 5/2) / sqrt(2*Pi).

A349581 G.f. A(x) satisfies: A(x) = 1 / (1 - 2x) + x (1 - 2x)^2 A(x)^4.

Original entry on oeis.org

1, 3, 12, 66, 460, 3681, 31848, 289176, 2714044, 26103468, 255876048, 2546717454, 25666830724, 261407935366, 2686191839232, 27815564456544, 289960011573212, 3040424427011492, 32046741183678288, 339345854532800136, 3608307717155678256, 38511520730570169033

Offset: 0

Ilya Gutkovskiy, Nov 22 2021

nonn

Second binomial transform of A002293.

Cf. A002293, A064613, A346646, A346762, A349582, A349584, A349590, A349591.

nmax = 21; A[] = 0; Do[A[x] = 1/(1 - 2 x) + x (1 - 2 x)^2 A[x]^4 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
Table[Sum[Binomial[n, k] Binomial[4 k, k] 2^(n - k)/(3 k + 1), {k, 0, n}], {n, 0, 21}]

a(n) = sum(k=0, n, binomial(n,k)*binomial(4*k,k)*2^(n-k)/(3*k+1)); \\ Michel Marcus, Nov 23 2021

a(n) = Sum_{k=0..n} binomial(n,k) * binomial(4*k,k) * 2^(n-k) / (3*k+1).
a(n) = 2^n*F([1/4, 1/2, 3/4, -n], [2/3, 1, 4/3], -2^7/3^3), where F is the generalized hypergeometric function. - Stefano Spezia, Nov 22 2021
a(n) ~ 2^(n - 10) * 155^(n + 3/2) / (sqrt(Pi) * n^(3/2) * 3^(3*n + 3/2)). - Vaclav Kotesovec, Nov 26 2021

A359643 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(4*k,k).

Original entry on oeis.org

1, 5, 37, 317, 2885, 27105, 259765, 2523813, 24768069, 244941833, 2437083697, 24367722725, 244639635749, 2464477467769, 24899468129405, 252202062544617, 2560119328830725, 26038134699958233, 265278657849511561, 2706809063101138409, 27657194997231516145, 282941098708193905485

Offset: 0

Vaclav Kotesovec, Jan 09 2023

nonn

In general, for m>1, Sum_{k=0..n} binomial(n,k) * binomial(m*k,k) ~ sqrt((m + (1 - 1/m)^(m-1))/(m-1)) * (1 + m^m/(m-1)^(m-1))^n / sqrt(2*Pi*n).

Robert Israel, Table of n, a(n) for n = 0..977

Cf. A026375, A188686.

Cf. A156887, A346646, A346664.

A359643 := proc(n)
    hypergeom([-n,1/4,1/2,3/4],[1/3,2/3,1],-256/27) ;
    simplify(%) ;
end proc:
seq(A359643(n),n=0..40) ; # R. J. Mathar, Jan 10 2023

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

a(n) = sum(k=0, n, binomial(n,k) * binomial(4*k,k)); \\ Michel Marcus, Jan 09 2023

a(n) ~ 283^(n + 1/2) / (2^(7/2) * sqrt(Pi*n) * 3^(3*n + 1/2)).
Conjecture D-finite with recurrence +81*n*(3*n-1)*(3*n-2)*a(n) +3*(243*n^3-8433*n^2+14984*n-7064)*a(n-1) +2*(-58607*n^3+297306*n^2-491401*n+269124)*a(n-2) +6*(n-2)*(56663*n^2-237722*n+252221)*a(n-3) -3*(n-2)*(n-3)*(111625*n-286402)*a(n-4) +110653*(n-2)*(n-3)*(n-4)*a(n-5)=0. - R. J. Mathar, Jan 09 2023
a(n) = 4F3( -n,1/4,1/2,3/4 ; 1/3, 2/3,1 ; -256/27). - R. J. Mathar, Jan 10 2023
a(n) = [x^n] (1 + 5*x + 6*x^2 + 4*x^3 + x^4)^n. - Ilya Gutkovskiy, Apr 17 2025

A381985 E.g.f. A(x) satisfies A(x) = exp(x) * B(xA(x)), where B(x) = 1 + xB(x)^3 is the g.f. of A001764.

Original entry on oeis.org

1, 2, 13, 217, 5937, 223641, 10725433, 625007993, 42883208609, 3386452550689, 302545287708201, 30170153462509545, 3322052185576104049, 400328811249634307249, 52406094009429908677049, 7405663486143907784247481, 1123601498350780798756198209, 182173718779147621454796872769

Offset: 0

Seiichi Manyama, Mar 11 2025

nonn

Cf. A381984, A381986.

Cf. A001764, A002293, A346646, A364987.

a(n) = n!*sum(k=0, n, (k+1)^(n-k)*binomial(4*k+1, k)/((4*k+1)*(n-k)!));

Let F(x) be the e.g.f. of A364987. F(x) = B(x*A(x)) = exp( 1/3 * Sum_{k>=1} binomial(3*k,k) * (x*A(x))^k/k ).

a(n) = n! * Sum_{k=0..n} (k+1)^(n-k) * A002293(k)/(n-k)!.

A381937 G.f. A(x) satisfies A(x) = (1 + x) * B(x*A(x)), where B(x) is the g.f. of A001764.

Original entry on oeis.org

1, 2, 6, 35, 240, 1805, 14386, 119365, 1020136, 8918423, 79380514, 716911887, 6553219720, 60513355786, 563648995020, 5289485238552, 49963186247220, 474655663418546, 4532279676629700, 43473774550929628, 418706702628897708, 4047555977981218963

Offset: 0

Seiichi Manyama, Mar 10 2025

nonn

Cf. A025227, A381787, A381940.

Cf. A001764, A346646, A365178.

a(n) = sum(k=0, n, binomial(4*k+1, k)*binomial(k+1, n-k)/(4*k+1));

a(n) = Sum_{k=0..n} binomial(4*k+1,k) * binomial(k+1,n-k)/(4*k+1).

a(n) = A365178(n) + A365178(n-1).

cp's OEIS Frontend

A378327 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(n*k,k) / ((n-1)*k + 1).

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A349581 G.f. A(x) satisfies: A(x) = 1 / (1 - 2*x) + x * (1 - 2*x)^2 * A(x)^4.

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A359643 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(4*k,k).

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A381985 E.g.f. A(x) satisfies A(x) = exp(x) * B(x*A(x)), where B(x) = 1 + x*B(x)^3 is the g.f. of A001764.

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Author

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PARI

Formula

A381937 G.f. A(x) satisfies A(x) = (1 + x) * B(x*A(x)), where B(x) is the g.f. of A001764.

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Author

Keywords

Crossrefs

Programs

PARI

Formula

A378327 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(nk,k) / ((n-1)k + 1).

A349581 G.f. A(x) satisfies: A(x) = 1 / (1 - 2x) + x (1 - 2x)^2 A(x)^4.

A381985 E.g.f. A(x) satisfies A(x) = exp(x) * B(xA(x)), where B(x) = 1 + xB(x)^3 is the g.f. of A001764.