A014330 -id:A014330 - cp's OEIS frontend

A204452 A014330 - A203577. Difference between the exponential convolution of A000108 (Catalan) with itself and the corresponding exponential half-convolution.

0, 1, 2, 11, 34, 212, 804, 5567, 24014, 178148, 839596, 6501420, 32658872, 259775440, 1368965576, 11080668871, 60613092662, 496461841956, 2798385807012, 23113333523180, 133539494791000, 1109722749130576, 6545965568001272

Offset: 0

Wolfdieter Lang, Jan 16 2012

For the exponential (also known as binomial) half-convolution of the Catalan sequence A000108 with itself see A203577.

			With A000108 = {1, 1, 2, 5, 14, 42,...}
  a(4) = 4*5*1 + 1*14*1 = 34.
  a(5) = 10*5*2 + 5*14*1 + 1*42*1 = 212.

Cf. A000108, A014330, A203577.

A204452 := proc(n)
    add( binomial(n,k)*A000108(k)*A000108(n-k), k=floor(n/2)+1..n) ;
end proc:
seq(A204452(n),n=0..50) ; # R. J. Mathar, Jul 27 2022

a(n) = sum(binomial(n,k)*C(k)*C(n-k),k=floor(n/2)+1..n), n>=0, with C(n)=A000108(n), the Catalan numbers.
E.g.f.: (C(x)^2 - C2(x^2))/2 with the e.g.f. C(x) of A000108, and the o.g.f. C2(x) of the sequence {(C(n)/n!)^2}. Compare this with the e.g.f. of A203577.
    C(x) = hypergeom([1/2],[2],4*x) (see the e.g.f. of A000108 for the version involving BesselI functions), and
    C2(x) = hypergeom([1/2,1/2],[1,2,2],16*x).

A014333 Three-fold exponential convolution of Catalan numbers with themselves.

Original entry on oeis.org

1, 3, 12, 57, 306, 1806, 11508, 78147, 559962, 4201038, 32792472, 264946446, 2206077804, 18860908644, 165050642736, 1474389557739, 13413397423482, 124030117316238, 1163661348170328, 11060842687616610, 106377560784576612, 1034009073326130876

Offset: 0

N. J. A. Sloane

nonn

G. C. Greubel, Table of n, a(n) for n = 0..930

Cf. A000108, A014330, A126869, A138364.

m:=40;
R:=PowerSeriesRing(Rationals(), m);
f:= func< x | (&+[(k+1-x)*x^(2*k)/(Factorial(k)*Factorial(k+1)): k in [0..m+2]]) >;
Coefficients(R!(Laplace( Exp(6*x)*( f(x) )^3 ))); // G. C. Greubel, Jan 06 2023

nmax = 20; CoefficientList[Series[E^(6*x)*(BesselI[0, 2*x] - BesselI[1, 2*x])^3, {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Nov 13 2017 *)

m=40
def f(x): return sum((k+1-x)*x^(2*k)/(factorial(k)*factorial(k+1)) for k in range(m+2))
def A014333_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( exp(6*x)*( f(x) )^3 ).egf_to_ogf().list()
A014333_list(m) # G. C. Greubel, Jan 06 2023

E.g.f.: exp(6*x)*(BesselI(0,2*x) - BesselI(1,2*x))^3. - Ilya Gutkovskiy, Nov 01 2017
From Vaclav Kotesovec, Nov 13 2017: (Start)
Recurrence: (n+1)*(n+2)*(n+3)*a(n) = 4*(6*n^3 + 13*n^2 + 2*n - 3)*a(n-1) - 4*(n-1)*(44*n^2 - 16*n - 21)*a(n-2) + 192*(n-2)*(n-1)*(2*n - 3)*a(n-3).
a(n) ~ 2^(2*n) * 3^(n + 9/2) / (Pi^(3/2) * n^(9/2)). (End)

A203577 Exponential (or binomial) half-convolution of the sequence A000108 (Catalan) with itself.

Original entry on oeis.org

1, 1, 4, 11, 58, 212, 1304, 5567, 37734, 178148, 1284124, 6501420, 48758648, 259775440, 2000594288, 11080668871, 86930955662, 496461841956, 3947716126292, 23113333523180, 185660199980696, 1109722749130576, 8983793097101144, 54645629076275356, 445109373450545608, 2748480598104423952

Offset: 0

Wolfdieter Lang, Jan 13 2012

For the definition of the exponential (also known as binomial) half-convolution of a sequence with itself see A203576, where also the rule for the e.g.f. is given.

			With Catalan = A000108 = {1, 1, 2, 5, 14, 42, ...}
a(4) = 1*1*14 + 4*1*5 + 6*2*2 = 58.
a(5) = 1*1*42 + 5*1*14 + 10*2*5 = 212.

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

Cf. A203576, A000108, A014330 (exponential convolution).

a[n_] := Sum[ Binomial[n, k]*CatalanNumber[k]*CatalanNumber[n - k], {k, 0, n/2}]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Jun 21 2013 *)

hat(b,n) = sum(k=0,n\2,binomial(n,k)*b(k)*b(n-k))
A203577(n)=hat(A000108,n)  \\ where A000108(n)=(2*n)!/n!/(n+1)! \\ - M. F. Hasler, Jan 13 2012

a(n) = Sum_{k=0..floor(n/2)} binomial(n,k)*Catalan(k)*Catalan(n-k), n >= 0.
E.g.f.: (C(x)^2 + C2(x^2))/2 with the e.g.f. C(x) of A000108, and the e.g.f. C2(x) := Sum_{n>=0} Catalan(n)^2*x^n/(n!)^2 of the scaled Catalan squares. See a comment above.
  C(x) = hypergeom([1/2],[2],4*x) (see A000108 for the version involving BesselI functions), and
  C2(x) = hypergeom([1/2,1/2],[1,2,2],16*x).
Recurrence: n*(n+1)^2 * (n+2)^2 * (3*n^6 - 39*n^5 + 166*n^4 - 322*n^3 + 316*n^2 - 153*n + 27)*a(n) = 12*(n-1)*n*(n+1)^2 * (3*n^7 - 34*n^6 + 113*n^5 - 121*n^4 - 19*n^3 + 68*n^2 + 17*n - 18)*a(n-1) + 32*(3*n^11 - 45*n^10 + 220*n^9 - 448*n^8 + 173*n^7 + 920*n^6 - 1696*n^5 + 842*n^4 + 580*n^3 - 846*n^2 + 360*n - 54)*a(n-2) - 768*(n-2)^3 * n *(3*n^7 - 34*n^6 + 113*n^5 - 121*n^4 - 19*n^3 + 68*n^2 + 17*n - 18)*a(n-3) + 2048*(n-3)^3 * (n-2)^2 * (3*n^6 - 21*n^5 + 16*n^4 + 12*n^3 + n^2 - 2)*a(n-4). - Vaclav Kotesovec, Feb 25 2014
a(n) ~ 2^(3*n+2)/(Pi*n^3) * (1 + (1+(-1)^n)/sqrt(2*Pi*n)). - Vaclav Kotesovec, Feb 25 2014

A277220 Exponential convolution of Fibonacci (A000045) and Catalan (A000108) numbers.

Original entry on oeis.org

0, 1, 3, 11, 43, 180, 790, 3590, 16745, 79705, 385615, 1890747, 9375216, 46931897, 236873261, 1204089630, 6159064015, 31678706490, 163739008070, 850051218980, 4430529313065, 23175017046351, 121617754070653, 640122809255716, 3378402106118508, 17875011275340275

Offset: 0

Vladimir Reshetnikov, Oct 06 2016

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A277251, A000045, A000108, A014330, A014334, A090826.

[(&+[Binomial(n,k)*Fibonacci(k)*Catalan(n-k): k in [0..n]]): n in [0..30]]; // G. C. Greubel, Oct 22 2018

Table[Sum[Binomial[n, k] Fibonacci[k] CatalanNumber[n - k], {k, 0, n}], {n, 0, 30}] (* or *)
Round@Table[(GoldenRatio^n Hypergeometric2F1[1/2, -n, 2, -4/GoldenRatio] - (-GoldenRatio)^(-n) Hypergeometric2F1[1/2, -n, 2, 4 GoldenRatio])/Sqrt[5], {n, 0, 30}] (* Round is equivalent to FullSimplify here, but is much faster *)

for(n=0, 30, print1(sum(k=0,n, binomial(n,k)*fibonacci(k)* binomial(2*n-2*k,n-k)/(n-k+1)), ", ")) \\ G. C. Greubel, Oct 22 2018

a(n) = Sum_{k=0..n} binomial(n,k) * A000045(k) * A000108(n-k).
a(n) = (phi^n * hypergeom([1/2, -n], [2], -4/phi) - (-phi)^(-n) * hypergeom([1/2, -n], [2], 4*phi))/sqrt(5), where phi = (1+sqrt(5))/2 = A001622.
Recurrence: 19*(n+1)*(n+2)*(11*n+13)*a(n) + 2*(55*n^3+208*n^2+311*n+230)*a(n+1) + 2*(55*n^3+373*n^2+674*n+206)*a(n+3) = (n+2)*(297*n^2+1022*n+617)*a(n+2) + (n+3)*(n+5)*(11*n+2)*a(n+4).
E.g.f.: 2*exp(5*x/2)*sinh(x*sqrt(5)/2)*(BesselI_0(2*x) - BesselI_1(2*x))/sqrt(5) (the product of e.g.f. for Fibonacci and Catalan numbers).
a(n) ~ (phi + 4)^(n + 3/2) / (8 * sqrt(5*Pi) * n^(3/2)), where phi = A001622 = (1 + sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Mar 10 2018

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.

A277251 Exponential convolution of Lucas (A000032) and Catalan (A000108) numbers.

Original entry on oeis.org

2, 3, 9, 29, 107, 430, 1840, 8230, 38015, 179873, 867079, 4242111, 21006358, 105072063, 530058079, 2693632580, 13775807415, 70847283680, 366167521240, 1900884870494, 9907318315587, 51822028122623, 271949090063769, 1431369293422604, 7554372307564282

Offset: 0

Vladimir Reshetnikov, Oct 06 2016

nonn

Cf. A277220, A000032, A000108, A014330, A014334, A090826.

Table[Sum[Binomial[n, k] LucasL[k] CatalanNumber[n - k], {k, 0, n}], {n, 0,
   30}] (* or *)
Round@Table[GoldenRatio^n Hypergeometric2F1[1/2, -n, 2, -4/GoldenRatio] + (-GoldenRatio)^(-n) Hypergeometric2F1[1/2, -n, 2, 4 GoldenRatio], {n, 0, 30}] (* Round is equivalent to FullSimplify here, but is much faster *)

a(n) = Sum_{k=0..n} binomial(n,k) * A000032(k) * A000108(n-k).
a(n) = phi^n * hypergeom([1/2, -n], [2], -4/phi) + (-phi)^(-n) * hypergeom([1/2, -n], [2], 4*phi), where phi = (1+sqrt(5))/2 = A001622.
Recurrence: 19*(n+1)*(n+2)*(11*n+13)*a(n) + 2*(55*n^3+208*n^2+311*n+230)*a(n+1) + 2*(55*n^3+373*n^2+674*n+206)*a(n+3) = (n+2)*(297*n^2+1022*n+617)*a(n+2) + (n+3)*(n+5)*(11*n+2)*a(n+4).
E.g.f.: 2*exp(5*x/2)*cosh(x*sqrt(5)/2)*(BesselI_0(2*x) - BesselI_1(2*x)) (the product of e.g.f. for Lucas and Catalan numbers).
a(n) ~ (phi + 4)^(n + 3/2) / (8 * sqrt(Pi) * n^(3/2)), where phi = A001622 = (1 + sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Mar 10 2018

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A204452 A014330 - A203577. Difference between the exponential convolution of A000108 (Catalan) with itself and the corresponding exponential half-convolution.

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A014333 Three-fold exponential convolution of Catalan numbers with themselves.

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A203577 Exponential (or binomial) half-convolution of the sequence A000108 (Catalan) with itself.

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A277220 Exponential convolution of Fibonacci (A000045) and Catalan (A000108) numbers.

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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(2*k*x)*(BesselI(0,2*x) - BesselI(1,2*x))^k.

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A277251 Exponential convolution of Lucas (A000032) and Catalan (A000108) numbers.

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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.