A024429 -id:A024429 - cp's OEIS frontend

A357572 Expansion of e.g.f. sinh(sqrt(3) * (exp(x)-1)) / sqrt(3).

0, 1, 1, 4, 19, 85, 406, 2191, 13105, 84190, 573121, 4127521, 31434184, 252388957, 2126998693, 18740283556, 172134162631, 1644920020417, 16324076578870, 167938152551491, 1787952325142341, 19667748794844550, 223217829954224029, 2610546296216999197

Offset: 0

Seiichi Manyama, Oct 05 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..562
Eric Weisstein's World of Mathematics, Bell Polynomial.

Cf. A024429, A264037, A357598.

Cf. A027710, A357615, A357737.

a(n) = sum(k=0, (n-1)\2, 3^k*stirling(n, 2*k+1, 2));

Bell_poly(n, x) = exp(-x)*suminf(k=0, k^n*x^k/k!);
a(n) = round((Bell_poly(n, sqrt(3))-Bell_poly(n, -sqrt(3)))/(2*sqrt(3)));

a(n) = Sum_{k=0..floor((n-1)/2)} 3^k * Stirling2(n,2*k+1).
a(n) = ( Bell_n(sqrt(3)) - Bell_n(-sqrt(3)) )/(2 * sqrt(3)), where Bell_n(x) is n-th Bell polynomial.
a(n) = 0; a(n) = Sum_{k=0..n-1} binomial(n-1, k) * A357615(k).

A088312 Number of sets of lists (cf. A000262) with even number of lists.

Original entry on oeis.org

1, 0, 1, 6, 37, 260, 2101, 19362, 201097, 2326536, 29668681, 413257790, 6238931821, 101415565836, 1765092183037, 32734873484250, 644215775792401, 13404753632014352, 293976795292186897, 6775966692145553526, 163735077313046119861, 4138498600079573989140

Offset: 0

Vladeta Jovovic, Nov 05 2003

From Peter Bala, Mar 27 2022: (Start)
a(2*n) is odd ; a(2*n+1) is even.
If k is odd then k*(k-1) divides a(k). Consequently, 6 divides a(6*n+3), 10 divides a(10*n+5), 14 divides a(14*n+7), and in general, if k is odd then 2*k divides a(2*k*n + k).
For a positive integer k, a(n+2*k) - a(n) is divisible by k. Thus the sequence obtained by taking a(n) modulo k is purely periodic with period 2*k. Calculation suggests that when k is even the exact period equals k, and when k is odd the exact period equals 2*k. (End)

Alois P. Heinz, Table of n, a(n) for n = 0..444
Peter Bala, Integer sequences that become periodic on reduction modulo k for all k
N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, arXiv:math/0509316 [math.NT], 2005-2006; J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.
N. J. A. Sloane, LAH transform

Cf. A000262, A001710, A027187, A024429, A024430, A088313, A111884.

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!(Laplace( Cosh(x/(1-x)) ))); // G. C. Greubel, Dec 13 2022

b:= proc(n, t) option remember; `if`(n=0, t, add(
      b(n-j, 1-t)*binomial(n-1, j-1)*j!, j=1..n))
    end:
a:= n-> b(n, 1):
seq(a(n), n=0..30);  # Alois P. Heinz, May 10 2016
A088312 := n -> ifelse(n=0, 1, (1/2)*(n - 1)*n!*hypergeom([1 - n/2, 3/2 - n/2], [3/2, 3/2, 2], 1/4)): seq(simplify(A088312(n)), n = 0..21); # Peter Luschny, Dec 14 2022

With[{m=30}, CoefficientList[Series[Cosh[x/(1-x)], {x,0,m}], x] * Range[0,m]!] (* Vaclav Kotesovec, Jul 04 2015 *)
Table[Sum[n!/(2*k)! Binomial[n - 1, 2*k - 1], {k, 0, Floor[n/2]}], {n, 0, 30}] (* Emanuele Munarini, Sep 03 2017 *)

def A088312_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( cosh(x/(1-x)) ).egf_to_ogf().list()
A088312_list(40) # G. C. Greubel, Dec 13 2022

E.g.f.: cosh(x/(1-x)).
a(n) = Sum_{k=1..floor(n/2)} n!/(2*k)!*binomial(n-1,2*k-1).
a(n) ~ 2^(-3/2) * n^(n-1/4) * exp(2*sqrt(n)-n-1/2). - Vaclav Kotesovec, Jul 04 2015
a(n+4) - 2*(2*n+5)*a(n+3) + (6*n^2+24*n+23)*a(n+2) - 2*(n+1)*(n+2)*(2*n+3)*a(n+1) + n*(n+1)^2*(n+2)*a(n) = 0. - Emanuele Munarini, Sep 03 2017
a(n) = (1/2)*(A000262(n) + (-1)^n*A111884(n)). - Peter Bala, Mar 27 2022
a(n) = (1/2)*(n-1)*n!*hypergeom([1 - n/2, 3/2 - n/2], [3/2, 3/2, 2], 1/4) for n >= 1. - Peter Luschny, Dec 14 2022

More terms from Vaclav Kotesovec, Jul 04 2015

a(0)-a(1) prepended by Alois P. Heinz, May 10 2016

A088313 Number of "sets of lists" (cf. A000262) with an odd number of lists.

Original entry on oeis.org

0, 1, 2, 7, 36, 241, 1950, 18271, 193256, 2270017, 29272410, 410815351, 6231230412, 101560835377, 1769925341366, 32838929702671, 646218442877520, 13441862819232001, 294656673023216946, 6788407001443004647, 163962850573039534580, 4142654439686285737201

Offset: 0

Vladeta Jovovic, Nov 05 2003

From Peter Bala, Mar 27 2022: (Start)
a(2*n) is even; in fact, 2*n*(2*n-1)*(2n-2) divides a(2*n). a(2*n+1) is odd.
For a positive integer k, a(n+2*k) - a(n) is divisible by k. Thus the sequence obtained by taking a(n) modulo k is purely periodic with period 2*k. Calculation suggests that when k is even the exact period equals k, and when k is odd the exact period equals 2*k. (End)

Alois P. Heinz, Table of n, a(n) for n = 0..444
N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, arXiv:math/0509316 [math.NT], 2005-2006; J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.
N. J. A. Sloane, LAH transform

Cf. A000262, A001710, A027187, A024429, A024430, A088312, A109777, A111884.

R:=PowerSeriesRing(Rationals(), 30); [0] cat Coefficients(R!(Laplace( Sinh(x/(1-x)) ))); // G. C. Greubel, Dec 13 2022

b:= proc(n, t) option remember; `if`(n=0, t, add(
      b(n-j, 1-t)*binomial(n-1, j-1)*j!, j=1..n))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..30);  # Alois P. Heinz, May 10 2016
A088313 := n -> ifelse(n=0, 0, n!*hypergeom([1/2 - n/2, 1 - n/2], [1/2, 1, 3/2], 1/4)): seq(simplify(A088313(n)), n = 0..21); # Peter Luschny, Dec 14 2022

With[{m=30}, CoefficientList[Series[Sinh[x/(1-x)], {x,0,m}], x] * Range[0,m]!] (* Vaclav Kotesovec, Jul 04 2015 *)

my(x='x+O('x^66)); concat(0, Vec(serlaplace(sinh(x/(1-x))))) \\ Joerg Arndt, Jul 16 2013

def A088313_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( sinh(x/(1-x)) ).egf_to_ogf().list()
A088313_list(40) # G. C. Greubel, Dec 13 2022

E.g.f.: sinh(x/(1-x)).
a(n) = Sum_{k=1..floor((n+1)/2)} n!/(2*k-1)!*binomial(n-1, 2*k-2).
E.g.f.: sinh(x/(1-x)) = x/(2-2*x)*E(0), where E(k)= 1 + 1/( 1 - x^2/(x^2 + 2*(1-x)^2*(k+1)*(2*k+3)/E(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jul 16 2013
a(n) ~ 2^(-3/2) * n^(n-1/4) * exp(2*sqrt(n)-n-1/2). - Vaclav Kotesovec, Jul 04 2015
a(n) = (1/2)*(A000262(n) - (-1)^n*A111884(n)). - Peter Bala, Mar 27 2022
a(n) = n!*hypergeom([1/2 - n/2, 1 - n/2], [1/2, 1, 3/2], 1/4) for n >= 1. - Peter Luschny, Dec 14 2022

a(0)=0 prepended by Alois P. Heinz, May 10 2016

A356572 Expansion of e.g.f. sinh( (exp(3*x) - 1)/3 ).

Original entry on oeis.org

0, 1, 3, 10, 45, 307, 2718, 26371, 265359, 2778976, 30916863, 372113623, 4873075056, 68908186765, 1037694932823, 16438615126282, 271972422548361, 4687666317874495, 84181305836224422, 1576083180118379527, 30757003280682603699, 624671260245315540568

Offset: 0

Seiichi Manyama, Oct 07 2022

nonn

Cf. A009599, A024429, A357617.

Cf. A357649.

With[{m = 20}, Range[0, m]! * CoefficientList[Series[Sinh[(Exp[3*x] - 1)/3], {x, 0, m}], x]] (* Amiram Eldar, Oct 07 2022 *)

my(N=30, x='x+O('x^N)); concat(0, Vec(serlaplace(sinh((exp(3*x)-1)/3))))

a(n) = sum(k=0, (n-1)\2, 3^(n-1-2*k)*stirling(n, 2*k+1, 2));

a(n) = Sum_{k=0..floor((n-1)/2)} 3^(n-1-2*k) * Stirling2(n,2*k+1).

a(n) ~ 3^n * exp(n/LambertW(3*n) - n - 1/3) * n^n / (LambertW(3*n)^n * 2*sqrt(1 + LambertW(3*n))). - Vaclav Kotesovec, Oct 07 2022

A121869 Monthly Problem 10791, first expression.

Original entry on oeis.org

-1, 1, 0, -5, -15, 104, 1827, 7893, -207000, -5646249, -47897675, 1479282600, 74711288407, 1396956334921, -21032523700672, -2719998717430365, -104158663871982343, -715846242343471272, 189941380201812700699, 14820744271258596866013, 507768838531742620183176

Offset: 0

N. J. A. Sloane, Sep 05 2006

sign

G. C. Greubel, Table of n, a(n) for n = 0..330
A. Fekete and G. Martin, Problem 10791: Squared Series Yielding Integers, Amer. Math. Monthly, 108 (No. 2, 2001), 177-178.

Cf. A121870, A024429, A024430, A121867, A121868.

List([0..25], n-> (-1)*Sum([0..n], k-> Stirling2(n,k)) *Sum([0..n], k-> (-1)^k*Stirling2(n,k)) ); # G. C. Greubel, Oct 08 2019

a:= func< n | (-1)*(&+[StirlingSecond(n,k): k in [0..n]])*(&+[ (-1)^k*StirlingSecond(n,k): k in [0..n]]) >;
[a(n): n in [0..25]]; // G. C. Greubel, Oct 08 2019

with(combinat): seq(-bell(n)*BellB(n, -1), n = 0..25); # G. C. Greubel, Oct 08 2019

Table[-BellB[n]*BellB[n, -1], {n,0,25}] (* G. C. Greubel, Oct 08 2019 *)

a(n) = (-1)*sum(k=0,n, stirling(n,k,2))*sum(k=0,n, (-1)^k*stirling(n,k,2));
vector(25, n, a(n-1)) \\ G. C. Greubel, Oct 08 2019

[ -sum(stirling_number2(n, k) for k in (0..n))*sum((-1)^k* stirling_number2(n,k) for k in (0..n)) for n in (0..25)] # G. C. Greubel, Oct 08 2019

a(n) = A024429(n)^2 - A024430(n)^2.

A357617 Expansion of e.g.f. sinh( (exp(4*x) - 1)/4 ).

Original entry on oeis.org

0, 1, 4, 17, 88, 657, 6844, 83393, 1072880, 14242785, 197046964, 2895895345, 45930435016, 789930042865, 14628150636012, 287915593953889, 5950831121362656, 128180962018224833, 2868724306984850020, 66704877850797014353, 1613138176448134032440

Offset: 0

Seiichi Manyama, Oct 07 2022

nonn

Cf. A009599, A024429, A356572.

Cf. A357650.

With[{m = 20}, Range[0, m]! * CoefficientList[Series[Sinh[(Exp[4*x] - 1)/4], {x, 0, m}], x]] (* Amiram Eldar, Oct 07 2022 *)

my(N=30, x='x+O('x^N)); concat(0, Vec(serlaplace(sinh((exp(4*x)-1)/4))))

a(n) = sum(k=0, (n-1)\2, 4^(n-1-2*k)*stirling(n, 2*k+1, 2));

a(n) = Sum_{k=0..floor((n-1)/2)} 4^(n-1-2*k) * Stirling2(n,2*k+1).

a(n) ~ 2^(2*n-1) * exp(n/LambertW(4*n) - n - 1/4) * n^n / (LambertW(4*n)^n * sqrt(1 + LambertW(4*n))). - Vaclav Kotesovec, Oct 07 2022

A357664 Expansion of e.g.f. sinh( (exp(2*x) - 1)/sqrt(2) )/sqrt(2).

Original entry on oeis.org

0, 1, 2, 6, 32, 220, 1592, 11944, 96000, 847120, 8209952, 86020704, 958326272, 11243157952, 138464594816, 1789358629504, 24250275913728, 344002396594432, 5092763802452480, 78443316497892864, 1253887341918199808, 20761127890765634560

Offset: 0

Seiichi Manyama, Oct 07 2022

nonn

Cf. A024429, A357665, A357666.

Cf. A009599, A264037, A357661.

my(N=30, x='x+O('x^N)); concat(0, apply(round, Vec(serlaplace(sinh((exp(2*x)-1)/sqrt(2))/sqrt(2)))))

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

a(n) = Sum_{k=0..floor((n-1)/2)} 2^(n-1-k) * Stirling2(n,2*k+1).

A357665 Expansion of e.g.f. sinh( (exp(3*x) - 1)/sqrt(3) )/sqrt(3).

Original entry on oeis.org

0, 1, 3, 12, 81, 765, 7938, 85239, 963819, 11801862, 158533443, 2320621569, 36425289816, 604576791405, 10532817901791, 192197187209484, 3673078679995677, 73486862051182425, 1536507360834633666, 33482575797899354235, 758209049155176114807

Offset: 0

Seiichi Manyama, Oct 07 2022

nonn

Cf. A024429, A357664, A357666.

Cf. A356572, A357572, A357662.

my(N=30, x='x+O('x^N)); concat(0, apply(round, Vec(serlaplace(sinh((exp(3*x)-1)/sqrt(3))/sqrt(3)))))

a(n) = sum(k=0, (n-1)\2, 3^(n-1-k)*stirling(n, 2*k+1, 2));

a(n) = Sum_{k=0..floor((n-1)/2)} 3^(n-1-k) * Stirling2(n,2*k+1).

A357666 Expansion of e.g.f. sinh( (exp(4*x) - 1)/2 )/2.

Original entry on oeis.org

0, 1, 4, 20, 160, 1872, 25024, 348224, 5055488, 78571776, 1332573184, 24695206912, 493816963072, 10492449771520, 234399640633344, 5480635606908928, 134015043318054912, 3427700843478056960, 91642829715498336256, 2556218693498006929408

Offset: 0

Seiichi Manyama, Oct 07 2022

nonn

Cf. A024429, A357664, A357665.

Cf. A357598, A357617, A357663.

my(N=20, x='x+O('x^N)); concat(0, Vec(serlaplace(sinh((exp(4*x)-1)/2)/2)))

a(n) = sum(k=0, (n-1)\2, 4^(n-1-k)*stirling(n, 2*k+1, 2));

a(n) = Sum_{k=0..floor((n-1)/2)} 4^(n-1-k) * Stirling2(n,2*k+1).

A358837 Number of odd-length multiset partitions of integer partitions of n.

Original entry on oeis.org

0, 1, 2, 4, 7, 14, 28, 54, 106, 208, 399, 757, 1424, 2642, 4860, 8851, 15991, 28673, 51095, 90454, 159306, 279067, 486598, 844514, 1459625, 2512227, 4307409, 7357347, 12522304, 21238683, 35903463, 60497684, 101625958, 170202949, 284238857, 473356564, 786196353

Offset: 0

Gus Wiseman, Dec 05 2022

nonn

			The a(1) = 1 through a(5) = 14 multiset partitions:
  {{1}}  {{2}}    {{3}}          {{4}}            {{5}}
         {{1,1}}  {{1,2}}        {{1,3}}          {{1,4}}
                  {{1,1,1}}      {{2,2}}          {{2,3}}
                  {{1},{1},{1}}  {{1,1,2}}        {{1,1,3}}
                                 {{1,1,1,1}}      {{1,2,2}}
                                 {{1},{1},{2}}    {{1,1,1,2}}
                                 {{1},{1},{1,1}}  {{1,1,1,1,1}}
                                                  {{1},{1},{3}}
                                                  {{1},{2},{2}}
                                                  {{1},{1},{1,2}}
                                                  {{1},{2},{1,1}}
                                                  {{1},{1},{1,1,1}}
                                                  {{1},{1,1},{1,1}}
                                                  {{1},{1},{1},{1},{1}}

Andrew Howroyd, Table of n, a(n) for n = 0..1000

The version for set partitions is A024429.
These multiset partitions are ranked by A026424.
The version for partitions is A027193.
The version for twice-partitions is A358824.
A001970 counts multiset partitions of integer partitions.
A063834 counts twice-partitions, strict A296122.
Cf. A000219, A141199, A336342, A358334, A358831.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
Table[Length[Select[Join@@mps/@Reverse/@IntegerPartitions[n],OddQ[Length[#]]&]],{n,0,10}]

P(v,y) = {1/prod(k=1, #v, (1 - y*x^k + O(x*x^#v))^v[k])}
seq(n) = {my(v=vector(n, k, numbpart(k))); (Vec(P(v,1)) - Vec(P(v,-1)))/2} \\ Andrew Howroyd, Dec 31 2022

G.f.: ((1/Product_{k>=1} (1-x^k)^A000041(k)) - (1/Product_{k>=1} (1+x^k)^A000041(k))) / 2. - Andrew Howroyd, Dec 31 2022

Terms a(11) and beyond from Andrew Howroyd, Dec 31 2022

cp's OEIS Frontend

A357572 Expansion of e.g.f. sinh(sqrt(3) * (exp(x)-1)) / sqrt(3).

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A088312 Number of sets of lists (cf. A000262) with even number of lists.

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A088313 Number of "sets of lists" (cf. A000262) with an odd number of lists.

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A356572 Expansion of e.g.f. sinh( (exp(3*x) - 1)/3 ).

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Mathematica

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A121869 Monthly Problem 10791, first expression.

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GAP

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A357617 Expansion of e.g.f. sinh( (exp(4*x) - 1)/4 ).

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A357664 Expansion of e.g.f. sinh( (exp(2*x) - 1)/sqrt(2) )/sqrt(2).

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