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 1-5 of 5 results.

A349603 a(n) = Sum_{k=0..n} binomial(n,k) * A000108(k) * k^(n-k).

Original entry on oeis.org

1, 1, 4, 20, 126, 937, 7938, 74909, 775022, 8688827, 104608026, 1342844846, 18273663268, 262347913479, 3957524475778, 62511713866200, 1030842278673510, 17700339693712731, 315740112103311666, 5839137279831300536, 111749137533005481700, 2209538389126578658875
Offset: 0

Views

Author

Vaclav Kotesovec, Nov 23 2021

Keywords

Crossrefs

Cf. A000248, A007317, A064613, A104455, A104498, A154623, A292632.

Programs

  • Mathematica
    Join[{1}, Table[Sum[Binomial[n, j]*CatalanNumber[j]*j^(n-j), {j, 0, n}], {n, 1, 25}]]
  • PARI
    a(n) = sum(k=0, n, binomial(n,k)*(binomial(2*k,k)/(k+1))*k^(n-k)); \\ Michel Marcus, Nov 23 2021

A349640 a(n) = Sum_{k=0..n} binomial(n,k) * A000108(k) * k!.

Original entry on oeis.org

1, 2, 7, 46, 485, 7066, 130987, 2946182, 77923561, 2369742130, 81467904431, 3124302688222, 132237820201357, 6123150708289226, 307903794151741075, 16709463201832993846, 973385368533058021457, 60583668821975488285282, 4012342371757905842648791, 281735471040327667890013070
Offset: 0

Views

Author

Vaclav Kotesovec, Nov 23 2021

Keywords

Comments

For each positive integer k, the sequence obtained by reducing a(n) modulo k is a periodic sequence with period dividing k. For example, modulo 5 the sequence becomes [1, 2, 2, 1, 0, 1, 2, 2, 1, 0, ...] with period 5. In particular, a(5*n+4) == 0 (mod 5). Cf. A047974. - Peter Bala, Mar 13 2025

Crossrefs

Cf. A007317, A064613, A292632, A349603, A349639, A224500.

Programs

  • Maple
    gf := exp(x)*(1 - sqrt(1 - 4*x))/(2*x): ser := series(gf, x, 24):
seq(n!*coeff(ser, x, n), n = 0..19);
# Alternative:
a := n -> `if`(n < 4, [1, 2, 7, 46][n + 1], ((4*n^2 - 12*n + 8)*a(n - 3) - (8*n^2 - 13*n + 5)*a(n - 2) + 4*n^2*a(n - 1))/(n + 1)):
seq(a(n), n = 0..19);  # Peter Luschny, Nov 23 2021
# Alternative
seq(simplify(hypergeom([-n, 1/2, 1], [2], -4)), n = 0..19); # Peter Bala, Mar 13 2025
  • Mathematica
    Table[Sum[Binomial[n, j]*CatalanNumber[j]*j!, {j, 0, n}], {n, 0, 20}]
  • PARI
    a(n) = sum(k=0, n, binomial(n,k) * (binomial(2*k,k)/(k+1)) * k!); \\ Michel Marcus, Nov 23 2021

Formula

a(n) ~ 2^(2*n + 1/2) * n^(n-1) / exp(n - 1/4).
From Peter Luschny, Nov 23 2021: (Start)
a(n) = n! * [x^n](exp(x)*(1 - sqrt(1 - 4*x))/(2*x)).
a(n) = (4*(n-1)*(n-2)*a(n - 3) - (n-1)*(8*n-5)*a(n - 2) + 4*n^2*a(n - 1))/(n + 1) for n >= 4.
a(n-1) = A224500(n) / n for n >= 1. (End)
a(n) = hypergeom([-n, 1/2, 1], [2], -4). - Peter Bala, Mar 13 2025

A292631 a(n) = n! * [x^n] exp(n*x)*(BesselI(0,2*x) + BesselI(1,2*x)).

Original entry on oeis.org

1, 2, 10, 75, 758, 9660, 148772, 2688420, 55784710, 1307378358, 34158527852, 984547901051, 31034429035260, 1062081192039140, 39218355263626632, 1554260970293874135, 65803396940022289734, 2964120950479432183950, 141548149894016562758300, 7143010414313948156920665, 379821534884560034711455956
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 20 2017

Keywords

Comments

The n-th term of the n-th binomial transform of A001405.

Links

Crossrefs

Main diagonal of A292630.
Cf. A001405, A186925, A292629, A292632.

Programs

  • Mathematica
    Table[n!*SeriesCoefficient[E^(n*x)*(BesselI[0,2*x] + BesselI[1,2*x]),{x,0,n}], {n,0,20}] (* Vaclav Kotesovec, Sep 20 2017 *)

Formula

a(n) = A292630(n,n).
a(n) ~ (BesselI(0,2) + BesselI(1,2)) * n^n. - Vaclav Kotesovec, Sep 20 2017

A293491 a(n) = n! * [x^n] exp((n+2)*x)*BesselI(0,2*x).

Original entry on oeis.org

1, 3, 18, 155, 1734, 23877, 390804, 7417377, 160256070, 3885021569, 104465601756, 3086353547433, 99399100528924, 3466411543407555, 130151205663179112, 5235127829223881895, 224609180728848273990, 10239557195235638377449, 494317596005491398892620, 25192788307121307053168673
Offset: 0

Views

Author

Ilya Gutkovskiy, Oct 10 2017

Keywords

Comments

The n-th term of the n-th binomial transform of A000984.

Links

Crossrefs

Cf. A000984, A026375, A081671, A098409, A098410, A104454, A186925, A292627, A292632.

Programs

  • Mathematica
    Table[n! SeriesCoefficient[Exp[(n + 2) x] BesselI[0, 2 x], {x, 0, n}], {n, 0, 19}]
Table[SeriesCoefficient[1/Sqrt[(1 - n x) (1 - (n + 4) x)], {x, 0, n}], {n, 0, 19}]
Join[{1}, Table[Sum[Binomial[n, k] Binomial[2 k, k] n^(n - k), {k, 0, n}], {n, 1, 19}]]
Table[(n + 2)^n HypergeometricPFQ[{1/2 - n/2, -n/2}, {1}, 4/(2 + n)^2], {n, 0, 19}]

Formula

a(n) = [x^n] 1/sqrt((1 - n*x)*(1 - (n + 4)*x)).
a(n) = Sum_{k=0..n} binomial(n,k)*binomial(2*k,k)*n^(n-k).
a(n) ~ exp(2) * BesselI(0,2) * n^n. - Vaclav Kotesovec, Oct 16 2017

A349639 a(n) = Sum_{k=0..n} binomial(n,k) * A000108(k) * k^k.

Original entry on oeis.org

1, 2, 11, 163, 4177, 150606, 7002679, 399296682, 26997867705, 2112814307980, 187919721166951, 18727570061711897, 2067435790679136937, 250474099952311886236, 33043529154916822685459, 4715582224589290429430011, 723854564711343436767660481, 118933484485939500023357177356
Offset: 0

Views

Author

Vaclav Kotesovec, Nov 23 2021

Keywords

Crossrefs

Cf. A007317, A064613, A291699, A292632, A349603, A349640.

Programs

  • Mathematica
    Table[1+Sum[Binomial[n, j]*CatalanNumber[j]*j^j, {j, 1, n}], {n, 0, 20}]
  • PARI
    a(n) = sum(k=0, n, binomial(n,k) * (binomial(2*k,k)/(k+1)) * k^k); \\ Michel Marcus, Nov 23 2021

Formula

a(n) ~ c * 2^(2*n) * n^(n - 3/2) /sqrt(Pi), where c = Sum_{k>=0} 1/(4^k*k!*exp(k)) = exp(exp(-1)/4) = 1.09633177846412646399584148732...
Showing 1-5 of 5 results.

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