A086672 -id:A086672 - cp's OEIS frontend

A064856 Stirling transform of Catalan numbers: a(n) = Sum_{k=0..n} stirling2(n,k)binomial(2k,k)/(k+1).

1, 1, 3, 12, 59, 338, 2185, 15613, 121553, 1020170, 9154963, 87276995, 879242215, 9319182044, 103537712361, 1201967382478, 14540040004755, 182840037042560, 2384985091689409, 32209645344213417, 449608555748234353, 6476887237235672388, 96156363230696213447

Offset: 0

Karol A. Penson, Oct 08 2001

Robert Israel, Table of n, a(n) for n = 0..400

Cf. A000108, A066053, A086672.

seq(add(Stirling2(n,k)*binomial(2*k,k)/(k+1),k=0..n), n=0..50); # Robert Israel, Sep 16 2016

Table[Sum[StirlingS2[n,k] Binomial[2k,k]/(k+1),{k,0,n}],{n,0,20}] (* Harvey P. Dale, Nov 01 2011 *)

{a(n)=polcoeff(sum(m=0, n, (2*m)!/(m!*(m+1)!)*x^m/prod(k=1, m, 1-k*x+x*O(x^n))), n)} /* Paul D. Hanna, Jul 20 2011 */

O.g.f.: Sum_{n>=1} C(2*n,n)/(n+1) * x^n / Product_{k=0..n} (1-k*x). - Paul D. Hanna, Jul 20 2011
E.g.f.: exp(2*exp(z)-2)*(BesselI(0, 2*exp(z)-2)-BesselI(1, 2*exp(z)-2)). Representation as a sum of an infinite series involving the confluent hypergeometric function 1F1, in Maple notation: a(n)=evalf(sum('k'^n*2^(2*'k')*GAMMA('k'+1/2)*evalf(hypergeom(['k'+1/2], ['k'+2], -4))/(sqrt(Pi)*'k'!*('k'+1)!), 'k'=0..infinity)), n=0, 1...
E.g.f.: hypergeom([1/2], [2], 4*(exp(x)-1)). - Vladeta Jovovic, Sep 11 2003

A201950 Central coefficients in Product_{k=0..n-1} (1 + k*x + x^2).

Original entry on oeis.org

1, 0, 2, 6, 28, 160, 1078, 8358, 73260, 716112, 7721844, 91039740, 1164932470, 16077368580, 238037983558, 3763371442530, 63276351409092, 1127406030014112, 21218146474666864, 420611921077524912, 8759617763834095796, 191208185756772875880

Offset: 0

Paul D. Hanna, Dec 06 2011

nonn

			The coefficients in Product_{k=0..n-1} (1+k*x+x^2) form triangle A201949:
(1);
1,(0), 1;
1, 1,(2), 1, 1;
1, 3, 5, (6), 5, 3, 1;
1, 6, 15, 24, (28), 24, 15, 6, 1;
1, 10, 40, 90, 139, (160), 139, 90, 40, 10, 1;
1, 15, 91, 300, 629, 945, (1078), 945, 629, 300, 91, 15, 1;
1, 21, 182, 861, 2520, 5019, 7377, (8358), 7377, 5019, 2520, 861, 182, 21, 1;
1, 28, 330, 2156, 8729, 23520, 45030, 65016, (73260), 65016, 45030, 23520, 8729, 2156, 330, 28, 1; ...
where coefficients in parenthesis form the initial terms of this sequence.

Vaclav Kotesovec, Table of n, a(n) for n = 0..300

Cf. A201949, A201951, A201952, A201953.

Cf. A086672, A324304 (variant).

Flatten[{1,Table[Coefficient[Expand[Product[1 + k*x + x^2,{k,0,n-1}]],x^n],{n,1,20}]}] (* Vaclav Kotesovec, Feb 10 2015 *)

{a(n) = polcoeff( prod(k=1,n,1+(k-1)*x+x^2+x*O(x^n)), n)}
for(n=0,30, print1(a(n),", "))

/* From series BesselI(0, 2*log(1 - x)), after Ilya Gutkovskiy */
{a(n) = n!*polcoeff( sum(m=0,n, log(1 - x +x*O(x^n))^(2*m)/m!^2), n)}
for(n=0,30, print1(a(n),", ")) \\ Paul D. Hanna, Feb 24 2019

Central terms of rows in irregular triangle A201949.
a(n) = (n-1)*a(n-1) + 2*A201952(n-1) for n>0. [corrected by Vaclav Kotesovec, May 04 2024]
E.g.f.: BesselI(0, 2*log(1 - x)). - Ilya Gutkovskiy, Feb 22 2019
E.g.f.: Sum_{n>=0} log(1 - x)^(2*n) / n!^2. [After Ilya Gutkovskiy - Paul D. Hanna, Feb 24 2019]

A306335 Expansion of e.g.f. BesselI(0,2log(1 + x)) + BesselI(1,2log(1 + x)).

Original entry on oeis.org

1, 1, 1, -1, 4, -21, 133, -981, 8244, -77694, 811194, -9292075, 115843000, -1561272571, 22618147199, -350481556959, 5784147674772, -101284047800632, 1875504207906184, -36616289396963678, 751702523788615816, -16187581390548113842, 364861626149143519378, -8590429045711448354359

Offset: 0

Ilya Gutkovskiy, Feb 08 2019

sign

Robert Israel, Table of n, a(n) for n = 0..400

Cf. A001405, A048994, A086672, A305560.

E:= BesselI(0,2*log(1 + x)) + BesselI(1,2*log(1 + x)):
S:= series(E,x,51):
seq(coeff(S,x,j)*j!,j=0..50); # Robert Israel, Feb 10 2019

nmax = 23; CoefficientList[Series[BesselI[0, 2 Log[1 + x]] + BesselI[1, 2 Log[1 + x]], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[StirlingS1[n, k] Binomial[k, Floor[k/2]], {k, 0, n}], {n, 0, 23}]

a(n) = sum(k=0, n, stirling(n, k, 1)*binomial(k, k\2)); \\ Michel Marcus, Feb 09 2019

a(n) = Sum_{k=0..n} Stirling1(n,k)*A001405(k).

A317169 Expansion of e.g.f. BesselI(1,2log(1 - x))/((1 - x)log(1 - x)).

Original entry on oeis.org

1, 1, 3, 12, 61, 375, 2699, 22232, 206086, 2122110, 24023623, 296474178, 3960532707, 56931074109, 876098828097, 14369369855760, 250215898045984, 4609913757678432, 89586669708676510, 1831372328505086980, 39284382532454768754, 882269612910279500214, 20703128006754726971507

Offset: 0

Ilya Gutkovskiy, Jul 23 2018

nonn

Cf. A001006, A086662, A086672, A317170.

a:=series(BesselI(1,2*log(1 - x))/((1 - x)*log(1 - x)), x=0, 23): seq(n!*coeff(a, x, n), n=0..22); # Paolo P. Lava, Mar 26 2019

nmax = 22; CoefficientList[Series[BesselI[1, 2 Log[1 - x]]/((1 - x) Log[1 - x]), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Abs[StirlingS1[n, k]] Hypergeometric2F1[(1 - k)/2, -k/2, 2, 4], {k, 0, n}], {n, 0, 22}]

a(n) = Sum_{k=0..n} |Stirling1(n,k)|*A001006(k).

A317170 Expansion of e.g.f. exp(exp(x) - 1)BesselI(1,2(exp(x) - 1))/(exp(x) - 1).

Original entry on oeis.org

1, 1, 3, 11, 48, 242, 1374, 8619, 58923, 434595, 3431263, 28817120, 256100717, 2397920319, 23567078396, 242343368931, 2600148486462, 29036252825090, 336754427112094, 4048299252733563, 50357053778129599, 647129716643654763, 8579133975080008700, 117178742009906802080, 1646975673395621229201

Offset: 0

Ilya Gutkovskiy, Jul 23 2018

nonn

Stirling transform of the Motzkin numbers (A001006).

N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Stirling Transform

Cf. A001006, A064856, A086672, A317169.

a:=series(exp(exp(x) - 1)*BesselI(1,2*(exp(x) - 1))/(exp(x) - 1), x=0, 26): seq(n!*coeff(a, x, n), n=0..24); # Paolo P. Lava, Mar 26 2019

nmax = 24; CoefficientList[Series[Exp[Exp[x] - 1] BesselI[1, 2 (Exp[x] - 1)]/(Exp[x] - 1), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[StirlingS2[n, k] Hypergeometric2F1[(1 - k)/2, -k/2, 2, 4], {k, 0, n}], {n, 0, 24}]

a(n) = Sum_{k=0..n} Stirling2(n,k)*A001006(k).

A355290 a(n) = Sum_{k=0..n} (-1)^(n-k) * Stirling2(n,k) * Catalan(k).

Original entry on oeis.org

1, 1, 1, 0, -3, -2, 23, 17, -333, 86, 6941, -17025, -160267, 1082864, 2273807, -56742606, 152154285, 2293098332, -22007462809, -15179437171, 1671107690083, -10716783889040, -58404948615167, 1439391012463810, -6701658223127029, -88340107011433060

Offset: 0

Seiichi Manyama, Jun 27 2022

sign

Cf. A000108, A006531, A064856, A086662, A086672.

A355290 := proc(n)
    add((-1)^(n-k)*stirling2(n,k)*A000108(k),k=0..n) ;
end proc:
seq(A355290(n),n=0..70) ; # R. J. Mathar, Mar 13 2023

a(n) = sum(k=0, n,(-1)^(n-k)*stirling(n, k, 2)*binomial(2*k, k)/(k+1));

my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, binomial(2*k, k)/(k+1)*x^k/prod(j=1, k, 1+j*x)))

G.f.: Sum_{k>=0} Catalan(k) * x^k / Product_{j=1..k} (1 + j*x).

A085573 2Sum(floor(C(n,w)/w),w=1..n/2-1)+floor(C(n,n/2)/(n/2)) if n is even, otherwise 2Sum(floor(C(n,w)/w),w=1..(n-1)/2).

Original entry on oeis.org

2, 6, 11, 20, 32, 56, 97, 172, 298, 534, 952, 1736, 3150, 5824, 10724, 20042, 37308, 70304, 131971, 250308, 473020, 901872, 1713596, 3281122, 6262254, 12033330, 23053047, 44431308, 85393280, 165008114, 318009610, 615878180, 1189803926, 2308781688

Offset: 2

N. J. A. Sloane, Jul 07 2003

nonn

Cf. A085568-A086672.

b := binomial; f3 := n->if n mod 2 = 0 then 2*add(floor(b(n,w)/w),w=1..n/2-1)+floor(b(n,n/2)/(n/2)); else 2*add(floor(b(n,w)/w),w=1..(n-1)/2); fi;

A317310 Expansion of e.g.f. (1 + x)^2BesselI(0,2log(1 + x)).

Original entry on oeis.org

1, 2, 4, 6, 4, 0, -2, 14, -100, 792, -6996, 68508, -737882, 8676200, -110627142, 1520662410, -22418697948, 352885526856, -5907074659016, 104782694989616, -1963418893492364, 38753471698684512, -803656781974363412, 17469671114170029708, -397223288562294817330, 9429329994809282773300

Offset: 0

Ilya Gutkovskiy, Jan 22 2019

sign

Cf. A000984, A048994, A086672, A305406.

a:=series((1 + x)^2*BesselI(0,2*log(1 + x)), x=0, 26): seq(n!*coeff(a, x, n), n=0..25); # Paolo P. Lava, Mar 26 2019

nmax = 25; CoefficientList[Series[(1 + x)^2 BesselI[0, 2 Log[1 + x]], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[StirlingS1[n, k] Binomial[2 k, k], {k, 0, n}], {n, 0, 25}]

my(x='x + O('x^30)); Vec(serlaplace((1 + x)^2*besseli(0,2*log(1 + x)))) \\ Michel Marcus, Mar 27 2019

a(n) = Sum_{k=0..n} Stirling1(n,k)*A000984(k).

cp's OEIS Frontend

A064856 Stirling transform of Catalan numbers: a(n) = Sum_{k=0..n} stirling2(n,k)*binomial(2*k,k)/(k+1).

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A201950 Central coefficients in Product_{k=0..n-1} (1 + k*x + x^2).

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A306335 Expansion of e.g.f. BesselI(0,2*log(1 + x)) + BesselI(1,2*log(1 + x)).

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A317169 Expansion of e.g.f. BesselI(1,2*log(1 - x))/((1 - x)*log(1 - x)).

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A317170 Expansion of e.g.f. exp(exp(x) - 1)*BesselI(1,2*(exp(x) - 1))/(exp(x) - 1).

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A355290 a(n) = Sum_{k=0..n} (-1)^(n-k) * Stirling2(n,k) * Catalan(k).

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A085573 2*Sum(floor(C(n,w)/w),w=1..n/2-1)+floor(C(n,n/2)/(n/2)) if n is even, otherwise 2*Sum(floor(C(n,w)/w),w=1..(n-1)/2).

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A317310 Expansion of e.g.f. (1 + x)^2*BesselI(0,2*log(1 + x)).

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A064856 Stirling transform of Catalan numbers: a(n) = Sum_{k=0..n} stirling2(n,k)binomial(2k,k)/(k+1).

A306335 Expansion of e.g.f. BesselI(0,2log(1 + x)) + BesselI(1,2log(1 + x)).

A317169 Expansion of e.g.f. BesselI(1,2log(1 - x))/((1 - x)log(1 - x)).

A317170 Expansion of e.g.f. exp(exp(x) - 1)BesselI(1,2(exp(x) - 1))/(exp(x) - 1).

A085573 2Sum(floor(C(n,w)/w),w=1..n/2-1)+floor(C(n,n/2)/(n/2)) if n is even, otherwise 2Sum(floor(C(n,w)/w),w=1..(n-1)/2).

A317310 Expansion of e.g.f. (1 + x)^2BesselI(0,2log(1 + x)).