A014333 -id:A014333 - cp's OEIS frontend

A014330 Exponential convolution of Catalan numbers with themselves.

1, 2, 6, 22, 92, 424, 2108, 11134, 61748, 356296, 2123720, 13002840, 81417520, 519550880, 3369559864, 22161337742, 147544048324, 992923683912, 6746101933304, 46226667046360, 319199694771696, 2219445498261152, 15529758665102416, 109291258152550712

Offset: 0

N. J. A. Sloane

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A000108, A014333, A053175, A126869, A138364.

A014330:= func< n | (&+[Binomial(n,k)*Catalan(k)*Catalan(n-k): k in [0..n]]) >;
[A014330(n): n in [0..40]]; // G. C. Greubel, Jan 06 2023

Table[Sum[Binomial[n, k]*Binomial[2*k, k]/(k+1)*Binomial[2*n-2*k, n-k]/(n-k+1),{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Feb 25 2014 *)

A014330(n)=sum(k=0,n,binomial(n,k)*A000108(k)*A000108(n-k))  \\ M. F. Hasler, Jan 13 2012

def c(n): return catalan_number(n)
def A014330(n): return sum( binomial(n,k)*c(k)*c(n-k) for k in range(n+1))
[A014330(n) for n in range(41)] # G. C. Greubel, Jan 06 2023

From Vladeta Jovovic, Jan 01 2004: (Start)
E.g.f.: exp(4*x)*(BesselI(0, 2*x) - BesselI(1, 2*x))^2.
a(n) = Sum_{k=0..n} binomial(n, k)*binomial(2*k, k)/(k+1)*binomial(2*n-2*k, n-k)/(n-k+1).
a(n) = 4^n*Sum_{k=0..n} (-4)^(-k)*binomial(n, k)*binomial(k, floor(k/2))*binomial(k+1, floor((k+1)/2)).
a(n) = binomial(2*n, n)/(n+1)*hypergeometric3F2([-n-1, -n, 1/2], [2, 1/2-n], -1). (End)
(n + 1)*(n + 2)*a(n) = 4*(3*n^2 + n - 1)*a(n - 1) - 32*(n - 1)^2*a(n - 2). - Vladeta Jovovic, Jul 15 2004
a(n) = Sum_{k=0..n} binomial(n,k)*A000108(k)*A000108(n-k). - Philippe Deléham, Aug 23 2006
a(n) = (4*A053175(n) - A053175(n+1)/4) / ((n+2)*2^n). - Mark van Hoeij, Jul 02 2010
G.f.: (1-6*x)*hypergeometric2F1([1/2, 1/2],[2],16*x^2/(4*x-1)^2)/(2*x*(4*x-1)) - x*(8*x-1)*hypergeometric2F1([3/2, 3/2],[3],16*x^2/(4*x-1)^2)/(4*x-1)^3 + 1/(2*x). - Mark van Hoeij, Oct 25 2011
E.g.f.: hypergeometric1F1([1/2],[2],4*x)^2, coinciding with the above given e.g.f. - Wolfdieter Lang, Jan 13 2012
a(n) ~ 8^(n+1) / (Pi*n^3). - Vaclav Kotesovec, Feb 25 2014

More terms from Vincenzo Librandi, Feb 27 2014

A294498 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. exp(2kx)(BesselI(0,2x) - BesselI(1,2*x))^k.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 6, 5, 0, 1, 4, 12, 22, 14, 0, 1, 5, 20, 57, 92, 42, 0, 1, 6, 30, 116, 306, 424, 132, 0, 1, 7, 42, 205, 752, 1806, 2108, 429, 0, 1, 8, 56, 330, 1550, 5328, 11508, 11134, 1430, 0, 1, 9, 72, 497, 2844, 12730, 40632, 78147, 61748, 4862, 0

Offset: 0

Ilya Gutkovskiy, Nov 01 2017

A(n,k) is the k-fold exponential convolution of Catalan numbers with themselves, evaluated at n.

			E.g.f. of column k: A_k(x) = 1 + k*x/1! +  k*(k + 1)*x^2/2! + k*(k^2 + 3*k + 1)*x^3/3! + k^2*(k^2 + 6*k + 7)*x^4/4! + k*(k^4 + 10*k^3 + 25*k^2 + 10*k - 4)*x^5/5! + ...
Square array begins:
  1,   1,    1,     1,     1,      1,  ...
  0,   1,    2,     3,     4,      5,  ...
  0,   2,    6,    12,    20,     30,  ...
  0,   5,   22,    57,   116,    205,  ...
  0,  14,   92,   306,   752,   1550,  ...
  0,  42,  424,  1806,  5328,  12730,  ...

Alois P. Heinz, Antidiagonals for n = 0..150, flattened

Columns k=0..3 give A000007, A000108, A014330, A014333.
Rows n=0..2 give  A000012, A001477, A002378.
Main diagonal gives A294511.
Cf. A009766, A033184.

C:= proc(n) option remember; binomial(2*n, n)/(n+1) end:
A:= proc(n, k) option remember; `if`(k=0, `if`(n=0, 1, 0), `if`(k=1, C(n),
      (h-> add(binomial(n, j)*A(j, h)*A(n-j, k-h), j=0..n))(iquo(k, 2))))
    end:
seq(seq(A(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, Jan 06 2023

Table[Function[k, n! SeriesCoefficient[Exp[2 k x] (BesselI[0, 2 x] - BesselI[1, 2 x])^k, {x, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten

E.g.f. of column k: exp(2*k*x)*(BesselI(0,2*x) - BesselI(1,2*x))^k.

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A014330 Exponential convolution of Catalan numbers with themselves.

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A294498 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. exp(2kx)(BesselI(0,2x) - BesselI(1,2*x))^k.

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cp's OEIS Frontend

A014330 Exponential convolution of Catalan numbers with themselves.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

SageMath

Formula

Extensions

A294498 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. exp(2*k*x)*(BesselI(0,2*x) - BesselI(1,2*x))^k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A294498 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. exp(2kx)(BesselI(0,2x) - BesselI(1,2*x))^k.