A024430 -id:A024430 - cp's OEIS frontend

A357293 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} Stirling2(n,k*j).

1, 1, 0, 1, 1, 0, 1, 0, 2, 0, 1, 0, 1, 5, 0, 1, 0, 0, 3, 15, 0, 1, 0, 0, 1, 8, 52, 0, 1, 0, 0, 0, 6, 25, 203, 0, 1, 0, 0, 0, 1, 25, 97, 877, 0, 1, 0, 0, 0, 0, 10, 91, 434, 4140, 0, 1, 0, 0, 0, 0, 1, 65, 322, 2095, 21147, 0, 1, 0, 0, 0, 0, 0, 15, 350, 1232, 10707, 115975, 0, 1, 0, 0, 0, 0, 0, 1, 140, 1702, 5672, 58194, 678570, 0

Offset: 0

Seiichi Manyama, Oct 17 2022

			Square array begins:
  1,   1,  1,  1,  1,  1, 1, ...
  0,   1,  0,  0,  0,  0, 0, ...
  0,   2,  1,  0,  0,  0, 0, ...
  0,   5,  3,  1,  0,  0, 0, ...
  0,  15,  8,  6,  1,  0, 0, ...
  0,  52, 25, 25, 10,  1, 0, ...
  0, 203, 97, 91, 65, 15, 1, ...

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (first 51 antidiagonals)
Eric Weisstein's World of Mathematics, Bell Polynomial.

Columns k=0-3 give: A000007, A000110, A024430, A143815.

Cf. A357119.

T(n, k) = sum(j=0, n, stirling(n, k*j, 2));

T(n, k) = if(k==0, 0^n, n!*polcoef(sum(j=0, n\k, (exp(x+x*O(x^n))-1)^(k*j)/(k*j)!), n));

Bell_poly(n, x) = exp(-x)*suminf(k=0, k^n*x^k/k!);
T(n, k) = if(k==0, 0^n, my(w=exp(2*Pi*I/k)); round(sum(j=0, k-1, Bell_poly(n, w^j)))/k);

For k > 0, e.g.f. of column k: Sum_{j>=0} (exp(x)-1)^(k*j)/(k*j)!.

For k > 0, T(n,k) = ( Sum_{j=0..k-1} Bell_n(w^j) )/k, where Bell_n(x) is n-th Bell polynomial and w = exp(2*Pi*i/k).

A357649 Expansion of e.g.f. cosh( (exp(3*x) - 1)/3 ).

Original entry on oeis.org

1, 0, 1, 9, 64, 435, 3097, 24822, 232759, 2517345, 30070954, 382827225, 5110770205, 71421582024, 1049487311485, 16286699945853, 267145966335088, 4616924929100535, 83622792656855125, 1578916985654901366, 30957723637379211115, 628927539690331202661

Offset: 0

Seiichi Manyama, Oct 07 2022

nonn

Cf. A009153, A024430, A357650.

Cf. A356572.

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

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

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

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

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.

A357650 Expansion of e.g.f. cosh( (exp(4*x) - 1)/4 ).

Original entry on oeis.org

1, 0, 1, 12, 113, 1000, 8977, 86996, 959905, 12303888, 179038689, 2840696540, 47684181393, 835731314808, 15277172343409, 292597596283684, 5900038421042753, 125488177929542944, 2809541905807203009, 65903118624174027436, 1610968753088423886257

Offset: 0

Seiichi Manyama, Oct 07 2022

nonn

Cf. A009153, A024430, A357649.

Cf. A357617.

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

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

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

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

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

A357661 Expansion of e.g.f. cosh( (exp(2*x) - 1)/sqrt(2) ).

Original entry on oeis.org

1, 0, 2, 12, 60, 320, 2040, 15568, 133648, 1230336, 11962400, 123144384, 1349008320, 15731096576, 194349866880, 2527082917120, 34392647418112, 488243791183872, 7216792525799936, 110936087161801728, 1771199461131500544, 29324602146652307456

Offset: 0

Seiichi Manyama, Oct 07 2022

nonn

Cf. A024430, A357662, A357663.

Cf. A009153, A264036, A357664.

With[{nn=30},CoefficientList[Series[Cosh[(Exp[2x]-1)/Sqrt[2]],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Mar 23 2025 *)

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

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

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

A357662 Expansion of e.g.f. cosh( (exp(3*x) - 1)/sqrt(3) ).

Original entry on oeis.org

1, 0, 3, 27, 198, 1485, 12825, 132678, 1582497, 20603727, 284290560, 4132840239, 63571690485, 1038868740000, 18022911716439, 330305863479615, 6355242571945878, 127721845479277737, 2672729031195365949, 58142565625982730462, 1313557910179640120061

Offset: 0

Seiichi Manyama, Oct 07 2022

nonn

Cf. A024430, A357661, A357663.

Cf. A357615, A357649, A357665.

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

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

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

A357663 Expansion of e.g.f. cosh( (exp(4*x) - 1)/2 ).

Original entry on oeis.org

1, 0, 4, 48, 464, 4480, 48448, 621824, 9320704, 154890240, 2746131456, 51237908480, 1007228375040, 20965557829632, 463091379159040, 10826828061147136, 266438312153120768, 6861616219559034880, 184128217520198123520, 5135753969867535941632

Offset: 0

Seiichi Manyama, Oct 07 2022

nonn

Cf. A024430, A357661, A357662.

Cf. A065143, A357650, A357666.

With[{nn=20},CoefficientList[Series[Cosh[(Exp[4x]-1)/2],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Aug 13 2025 *)

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

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

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

A357667 Expansion of e.g.f. cosh( 3 * (exp(x) - 1) ).

Original entry on oeis.org

1, 0, 9, 27, 144, 945, 6273, 44226, 339399, 2796795, 24387786, 223853355, 2159078445, 21827316888, 230536050165, 2536213188519, 28994911890048, 343806474384045, 4220933769308205, 53566838971016418, 701650841036287275, 9473067208871584407

Offset: 0

Seiichi Manyama, Oct 08 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..547
Eric Weisstein's World of Mathematics, Bell Polynomial.

Column k=9 of A357681.
Cf. A024430, A065143, A264036, A357615.
Cf. A027710, A357649, A357668.

my(x='x+O('x^30)); Vec(serlaplace(cosh(3*(exp(x)-1))))

a(n) = sum(k=0, n\2, 9^k*stirling(n, 2*k, 2));

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

E.g.f.: cosh( 3 * (exp(x) - 1) ).
a(n) = Sum_{k=0..floor(n/2)} 9^k * Stirling2(n,2*k).
a(n) = ( Bell_n(3) + Bell_n(-3) )/2, where Bell_n(x) is n-th Bell polynomial.
a(n) = 1; a(n) = 9 * Sum_{k=0..n-1} binomial(n-1, k) * A357668(k).

A096648 Number of partitions of an n-set with odd number of even blocks.

Original entry on oeis.org

0, 1, 3, 7, 25, 106, 434, 2045, 10707, 57781, 338195, 2115664, 13796952, 95394573, 692462671, 5235101739, 41436754261, 341177640610, 2915100624274, 25866987547865, 237448494222575, 2252995117706961, 22078799199129799, 222971522853648704, 2319210969809731600

Offset: 1

Vladeta Jovovic, Aug 14 2004

Alois P. Heinz, Table of n, a(n) for n = 1..500

Cf. A000701.

Cf. A024429, A024430, A096647.

with(combinat):
b:= proc(n, i, t) option remember; `if`(n=0, t, `if`(i<1,
      0, add(multinomial(n, n-i*j, i$j)/j!*b(n-i*j, i-1,
      irem(t+`if`(irem(i, 2)=0, j, 0), 2)), j=0..n/i)))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=1..30);  # Alois P. Heinz, Mar 08 2015

multinomial[n_, k_List] := n!/Times @@ (k!); b[n_, i_, t_] := b[n, i, t] = If[n == 0, t, If[i<1, 0, Sum[multinomial[n, Join[{n-i*j}, Array[i&, j]]]/j!*b[n-i*j, i-1, Mod[t+If[Mod[i, 2] == 0, j, 0], 2]], {j, 0, n/i}]]]; a[n_] := b[n, n, 2]; Table[ a[n], {n, 1, 30}] (* Jean-François Alcover, May 13 2015, after Alois P. Heinz *)
With[{nn=30},Rest[CoefficientList[Series[Exp[Sinh[x]]Sinh[Cosh[x]-1], {x,0,nn}],x] Range[0,nn]!]] (* Harvey P. Dale, Sep 03 2016 *)

E.g.f.: exp(sinh(x))*sinh(cosh(x)-1).
a(2*n) = A024429(2*n) and a(2*n+1) = A024430(2*n+1). - Jonathan Vos Post, Oct 19 2005
a(n) = sum{k=0..n, if(mod(n-k,2)=1, A048993(n,k), 0)}. - Paul Barry, May 19 2006

More terms from Emeric Deutsch, Nov 16 2004

A121870 Monthly Problem 10791, second expression.

Original entry on oeis.org

1, 1, 2, 9, 61, 554, 6565, 96677, 1716730, 36072181, 881242577, 24674241834, 783024550969, 27896201305769, 1106485798248706, 48517267642373105, 2337333266369553253, 123040664089658462650, 7043260281573138384701, 436533086101058798529933

Offset: 0

N. J. A. Sloane, Sep 05 2006

nonn

G. C. Greubel, Table of n, a(n) for n = 0..325
A. Fekete and G. Martin, Problem 10791: Squared Series Yielding Integers, Amer. Math. Monthly, 108 (No. 2, 2001), 177-178.

Cf. A024429, A024430, A121867, A121868, A121869.

List([0..25], n-> (Sum([0..Int(n/2)], k-> Stirling2(n,2*k)*(-1)^(k)) )^2 + (Sum([0..Int(n/2)], k-> (-1)^k*Stirling2(n,2*k+1)))^2 ); # G. C. Greubel, Oct 08 2019

C:= ComplexField(); a:= func< n | Round(Abs( (&+[I^k*StirlingSecond(n,k): k in [0..n]])^2 )) >;
[a(n): n in [0..25]]; // G. C. Greubel, Oct 08 2019

Maple

A121870a:= proc(a) local i, t: i:=1: t:=0: for i to 100 do t:=t+evalf((i^(a-1))*(I)^i/(i)!): od: RETURN(round(abs(t^2))): end: a:= A121870a(n); # Russell Walsmith, Apr 18 2008 # Alternate: seq(abs(BellB(n,I))^2, n=0..30); # Robert Israel, Oct 15 2017

Mathematica

Table[Abs[BellB[n, I]]^2, {n, 0, 20}] (* Vladimir Reshetnikov, Oct 15 2017 *)

PARI

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

Sage

[abs( sum(I^k*stirling_number2(n,k) for k in (0..n))^2 ) for n in (0..25)] # G. C. Greubel, Oct 08 2019

Formula

a(n) = A121867(n)^2 + A121868(n)^2.

From Gary W. Adamson, Jul 22 2011: (Start)

sqrt(a(n)) = upper left term in M^n as to the modulus of a polar term; M = an infinite square production matrix in which a column of (i, i, i, ...) is appended to the right of Pascal's triangle, as follows (with i = sqrt(-1)):

1, i, 0, 0, 0, ...

1, 1, i, 0, 0, ...

1, 2, 1, i, 0, ...

1, 3, 3, 1, i, ...

... (End)

a(n) = |B_n(i)|^2, where B_n(x) is the n-th Bell polynomial, i = sqrt(-1) is the imaginary unit. - Vladimir Reshetnikov, Oct 15 2017

a(n) ~ (n*exp(-1 + Re(LambertW(i*n)) / Abs(LambertW(i*n))^2) / Abs(LambertW(i*n)))^(2*n) / Abs(1 + LambertW(i*n)), where i is the imaginary unit. - Vaclav Kotesovec, Jul 28 2021

cp's OEIS Frontend

A357293 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} Stirling2(n,k*j).

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A357649 Expansion of e.g.f. cosh( (exp(3*x) - 1)/3 ).

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A121869 Monthly Problem 10791, first expression.

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A357650 Expansion of e.g.f. cosh( (exp(4*x) - 1)/4 ).

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A357661 Expansion of e.g.f. cosh( (exp(2*x) - 1)/sqrt(2) ).

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A357662 Expansion of e.g.f. cosh( (exp(3*x) - 1)/sqrt(3) ).

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PARI

PARI

Formula

A357663 Expansion of e.g.f. cosh( (exp(4*x) - 1)/2 ).

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Mathematica

PARI

PARI

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A357667 Expansion of e.g.f. cosh( 3 * (exp(x) - 1) ).

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