cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A361278 Expansion of e.g.f. exp(x * (1+x)^2).

Original entry on oeis.org

1, 1, 5, 19, 97, 581, 3661, 26335, 202049, 1659817, 14621941, 135567851, 1326672865, 13624218349, 146056961597, 1633376573431, 18980051829121, 228677164878545, 2852155973178469, 36740599423566787, 488127224550517601, 6678832987859315221
Offset: 0

Views

Author

Seiichi Manyama, Mar 06 2023

Keywords

Links

Crossrefs

Column k=2 of A361277.
Cf. A082579.

Programs

  • Maple
    A361278 := proc(n)
    n!*add(binomial(2*k,n-k)/k!,k=0..n) ;
end proc:
seq(A361278(n),n=0..60) ; #R. J. Mathar, Mar 12 2023
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x*(1+x)^2)))
  • PARI
    a(n) = n!*sum(k=0, n, binomial(2*k, n-k)/k!);
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!*sum(j=1, i, j*binomial(2, j-1)*v[i-j+1]/(i-j)!)); v;

Formula

a(n) = n! * Sum_{k=0..n} binomial(2*k,n-k)/k!.
a(0) = 1; a(n) = (n-1)! * Sum_{k=1..n} k * binomial(2,k-1) * a(n-k)/(n-k)!.
D-finite with recurrence a(n) -a(n-1) +4*(-n+1)*a(n-2) -3*(n-1)*(n-2)*a(n-3)=0. - R. J. Mathar, Mar 12 2023
a(n) ~ 3^(n/3 - 1/2) * n^(2*n/3) / exp(2*n/3 - 2*3^(-2/3)*n^(2/3) - 3^(-7/3)*n^(1/3) + 4/81) * (1 + 953*3^(1/3)/(4374*n^(1/3)) - 2051059*3^(2/3)/(191318760*n^(2/3))). - Vaclav Kotesovec, Nov 11 2023

A361279 Expansion of e.g.f. exp(x * (1+x)^3).

Original entry on oeis.org

1, 1, 7, 37, 241, 1981, 17551, 171697, 1860097, 21609721, 268697431, 3566446621, 50060084977, 740156116597, 11496472967071, 186824483634601, 3167058238988161, 55882288483846897, 1023891003620741287, 19440027237549627541, 381822392009503555441
Offset: 0

Views

Author

Seiichi Manyama, Mar 06 2023

Keywords

Links

Crossrefs

Column k=3 of A361277.
Cf. A091695.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x*(1+x)^3)))
  • PARI
    a(n) = n!*sum(k=0, n, binomial(3*k, n-k)/k!);
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!*sum(j=1, i, j*binomial(3, j-1)*v[i-j+1]/(i-j)!)); v;

Formula

a(n) = n! * Sum_{k=0..n} binomial(3*k,n-k)/k!.
a(0) = 1; a(n) = (n-1)! * Sum_{k=1..n} k * binomial(3,k-1) * a(n-k)/(n-k)!.
D-finite with recurrence a(n) -a(n-1) +6*(-n+1)*a(n-2) -9*(n-1)*(n-2)*a(n-3) -4*(n-1)*(n-2)*(n-3)*a(n-4)=0. - R. J. Mathar, Mar 13 2023
a(n) ~ 2^(n/2 - 1) * n^(3*n/4) / exp(3*n/4 - 3*n^(3/4)/2^(3/2) - 15*n^(1/2)/64 + n^(1/4)/2^(19/2) + 27/1024) * (1 + 724053*sqrt(2)/(2621440*n^(1/4))). - Vaclav Kotesovec, Nov 11 2023

A361281 a(n) = n! * Sum_{k=0..n} binomial(n*k,n-k)/k!.

Original entry on oeis.org

1, 1, 5, 37, 481, 10001, 288901, 10820965, 511186817, 29843419681, 2106779832901, 176180844038981, 17165338119936865, 1924030148121500017, 245630480526435293381, 35409038825312233143301, 5719025066628373334423041, 1027649751647068260334391105
Offset: 0

Views

Author

Seiichi Manyama, Mar 06 2023

Keywords

Comments

From Peter Bala, Mar 12 2023: (Start)
It appears that a(n) == 1 (mod 4) and a(5*n+2) == 0 (mod 5) for all n. More generally we conjecture that a(n+k) == a(n) (mod k) for all n and k. If true, then for each k, the sequence a(n) taken modulo k is a periodic sequence and the period divides k.
Let F(x) and G(x) be power series with integer coefficients with G(0) = 1. Define b(n) = n! * [x^n] F(x)*exp(x*G(x)^n). Then we conjecture that b(n+k) == b(n) (mod k) for all n and k. The present sequence is the case F(x) = 1, G(x) = 1 + x. Cf. A278070. (End)

Links

Crossrefs

Main diagonal of A361277.
Cf. A096131, A099237, A226391, A278070, A293013.

Programs

  • PARI
    a(n) = n!*sum(k=0, n, binomial(n*k, n-k)/k!);

Formula

a(n) = n! * [x^n] exp(x * (1+x)^n).
log(a(n)) ~ n*(2*log(n) - log(log(n)) - 1 - log(2) + log(log(n))/log(n) + 1/(2*log(n)) + log(2)/log(n) - 1/(8*log(n)^2)). - Vaclav Kotesovec, Mar 12 2023

A361280 Expansion of e.g.f. exp(x * (1+x)^4).

Original entry on oeis.org

1, 1, 9, 61, 481, 4881, 55321, 682669, 9343041, 139078081, 2216425321, 37736834301, 683184324769, 13064452686481, 262867726142841, 5549111222344621, 122499654278797441, 2819926900630750209, 67539541277010100681, 1679557316488693881661
Offset: 0

Views

Author

Seiichi Manyama, Mar 06 2023

Keywords

Links

Crossrefs

Column k=4 of A361277.
Cf. A361283.

Programs

  • Maple
    A361280 := proc(n)
    n!*add(binomial(4*k,n-k)/k!,k=0..n) ;
end proc:
seq(A361280(n),n=0..60) ; # R. J. Mathar, Mar 12 2023
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x*(1+x)^4)))
  • PARI
    a(n) = n!*sum(k=0, n, binomial(4*k, n-k)/k!);
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!*sum(j=1, i, j*binomial(4, j-1)*v[i-j+1]/(i-j)!)); v;

Formula

a(n) = n! * Sum_{k=0..n} binomial(4*k,n-k)/k!.
a(0) = 1; a(n) = (n-1)! * Sum_{k=1..n} k * binomial(4,k-1) * a(n-k)/(n-k)!.
D-finite with recurrence a(n) -a(n-1) +8*(-n+1)*a(n-2) -18*(n-1)*(n-2)*a(n-3) -16*(n-1)*(n-2)*(n-3)*a(n-4) -5*(n-1)*(n-2)*(n-3)*(n-4)*a(n-5)=0. - R. J. Mathar, Mar 12 2023
a(n) ~ 5^(n/5 - 1/2) * n^(4*n/5) * exp(-256/15625 - 249*5^(4/5)*n^(1/5)/78125 + 236*5^(3/5)*n^(2/5)/9375 + 22*5^(2/5)*n^(3/5)/125 + 4*5^(-4/5)*n^(4/5) - 4*n/5) * (1 + 15409886*5^(1/5)/(48828125*n^(1/5))). - Vaclav Kotesovec, Nov 11 2023
Showing 1-4 of 4 results.

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