A005046 -id:A005046 - cp's OEIS frontend

A097322 (2n)! divided by denominator of Taylor expansion of exp(cos(x)-1).

1, 4, 1, 1, 4, 1, 1, 4, 1, 11, 4, 1, 1, 4, 11, 1, 4, 31, 1, 44, 29, 1, 4, 1, 11, 4, 29, 1, 4, 11, 1, 4, 31, 1, 1276, 1, 41, 4, 1, 11, 116, 1, 1, 292, 11, 1, 4, 31, 127571, 44, 1, 1, 4, 1, 319, 4, 41, 1, 4, 11, 1, 4, 899, 1, 44, 1, 1, 4, 29, 11, 4, 1, 1, 4, 583, 1, 4756, 361429

Offset: 1

Ralf Stephan, Aug 08 2004

Equals A010050(n) / A047690(n).

Cf. A005046.

a(n)=(2*n)!/denominator(polcoeff(Ser(exp(cos(x)-1)),2*n))

A327005 T(n, k) = Sum_{i=1..n} BM[k][i] where BM is the BellMatrix(x -> x mod n) as defined in A264428. Square array read by ascending antidiagonals for n >= 1 and k >= 1.

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 2, 4, 0, 1, 0, 1, 2, 3, 0, 0, 1, 0, 1, 2, 6, 21, 31, 0, 1, 0, 1, 2, 6, 20, 57, 0, 0, 1, 0, 1, 2, 6, 24, 101, 231, 379, 0, 1, 0, 1, 2, 6, 24, 100, 422, 1394, 0, 0, 1, 0, 1, 2, 6, 24, 105, 505, 2201, 5476, 6556, 0

Offset: 1

Peter Luschny, Aug 13 2019

Rows converge to the main diagonal A327006.

			[1] 1, 0, 0, 0, 0,  0,   0,   0,    0,     0,      0,      0, ...
[2] 1, 0, 1, 0, 4,  0,  31,   0,  379,     0,   6556,      0, ...
[3] 1, 0, 1, 2, 3, 21,  57, 231, 1394,  5476,  32616, 203105, ...
[4] 1, 0, 1, 2, 6, 20, 101, 422, 2201, 12560,  76846, 483892, ...
[5] 1, 0, 1, 2, 6, 24, 100, 505, 2620, 15383,  97480, 657305, ...
[6] 1, 0, 1, 2, 6, 24, 105, 504, 2759, 16186, 103494, 710384, ...

Peter Luschny, The Bell transform.
J. Riordan, Letter, Jul 06 1978.

A005046 is a bisection of row 2. Main diagonal is A327006.

Cf. A264428.

# BellMatrix is defined in A264428.
T := proc(n, k) BellMatrix(x -> modp(x, n), k): add(i, i in %[k]) end:
seq(seq(T(n-k+1,k), k=1..n), n=1..12);

A330041 Expansion of e.g.f. exp(cosh(exp(x) - 1) - 1).

Original entry on oeis.org

1, 0, 1, 3, 11, 55, 322, 2114, 15556, 127005, 1135374, 11011220, 115080825, 1288589757, 15379512670, 194796087841, 2608470709562, 36805935282625, 545626818921885, 8475730766054047, 137637670315066835, 2331584745107027528, 41122505417366272200

Offset: 0

Ilya Gutkovskiy, Nov 28 2019

nonn

Stirling transform of A005046 (with interpolated zeros).

Exponential transform of A024430.

Alois P. Heinz, Table of n, a(n) for n = 0..485

Cf. A000258, A005046, A024430, A185369, A330021.

g:= proc(n) option remember; `if`(n=0, 1, add(
      binomial(2*n-1, 2*k-1) *g(n-k), k=1..n))
    end:
b:= proc(n, m) option remember; `if`(n=0,
     `if`(m::odd, 0, g(m/2)), m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..22);  # Alois P. Heinz, Jun 23 2023

nmax = 22; CoefficientList[Series[Exp[Cosh[Exp[x] - 1] - 1], {x, 0, nmax}], x] Range[0, nmax]!

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1) * A024430(k) * a(n-k).

A352145 Expansion of e.g.f. exp(-1 + cos(x) + sin(x)).

Original entry on oeis.org

1, 1, 0, -3, -5, 12, 71, 7, -1028, -2573, 14793, 100188, -128831, -3445791, -5741800, 113954461, 601512787, -3296210612, -41316895641, 37322755431, 2570678600548, 6983413204755, -149303353515823, -1080122148248420, 7405149869523649, 119115584584019713

Offset: 0

Seiichi Manyama, Mar 15 2022

sign

Seiichi Manyama, Table of n, a(n) for n = 0..586

Cf. A002017, A005046, A292935, A352377.

my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(-1+cos(x)+sin(x))))

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

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

A352465 a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} binomial(2n,2k)^2 * k * a(n-k).

Original entry on oeis.org

1, 1, 19, 1576, 356035, 172499176, 154989443170, 234120771123513, 553941959716031715, 1945912976888526218512, 9731900583801946493234794, 66990924607889809703423378253, 617312916540194845307221190273098, 7439659538258619452171059589120614701

Offset: 0

Ilya Gutkovskiy, Mar 17 2022

nonn

Cf. A005046, A023998, A346220, A352466.

a[0] = 1; a[n_] := a[n] = (1/n) Sum[Binomial[2 n, 2 k]^2 k a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 13}]
nmax = 26; Take[CoefficientList[Series[Exp[Sum[x^(2 k)/(2 k)!^2, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!^2, {1, -1, 2}]

Sum_{n>=0} a(n) * x^(2*n) / (2*n)!^2 = exp( Sum_{n>=1} x^(2*n) / (2*n)!^2 ).

Sum_{n>=0} a(n) * x^(2*n) / (2*n)!^2 = exp( (BesselI(0,2*sqrt(x)) + BesselJ(0,2*sqrt(x))) / 2 - 1 ).

A363073 Number of set partitions of [n] such that each element is contained in a block whose block size parity coincides with the parity of the element.

Original entry on oeis.org

1, 1, 0, 0, 1, 2, 0, 0, 20, 48, 0, 0, 1147, 3968, 0, 0, 173203, 709488, 0, 0, 53555964, 246505600, 0, 0, 28368601065, 148963383616, 0, 0, 24044155851601, 141410718244864, 0, 0, 30934515698084780, 198914201874983936, 0, 0, 57215369885233295955, 398742900995358584320

Offset: 0

Alois P. Heinz, May 17 2023

nonn

All odd elements are in blocks with an odd block size and all even elements are in blocks with an even block size.

			a(0) = 1: (), the empty partition.
a(1) = 1: 1.
a(4) = 1: 1|24|3.
a(5) = 2: 135|24, 1|24|3|5.
a(8) = 20: 135|2468|7, 135|24|68|7, 137|2468|5, 137|24|5|68, 135|26|48|7, 135|28|46|7, 137|26|48|5, 137|28|46|5, 157|2468|3, 157|24|3|68, 1|2468|357, 1|24|357|68, 1|2468|3|5|7, 1|24|3|5|68|7, 157|26|3|48, 157|28|3|46, 1|26|357|48, 1|28|357|46, 1|26|3|48|5|7, 1|28|3|46|5|7.

Alois P. Heinz, Table of n, a(n) for n = 0..672
Wikipedia, Partition of a set

Cf. A000110, A003724, A005046, A124419, A274538, A275679.

b:= proc(n, t) option remember; `if`(n=0, 1, add(
     `if`((j+t)::even, b(n-j, t)*binomial(n-1, j-1), 0), j=1..n))
    end:
a:= n-> (h-> b(n-h, 1)*b(h, 0))(iquo(n, 2)):
seq(a(n), n=0..40);

b[n_, t_] := b[n, t] = If[n == 0, 1, Sum[If[EvenQ[j + t], b[n - j, t]* Binomial[n - 1, j - 1], 0], {j, 1, n}]];
a[n_] := b[n - #, 1]*b[#, 0]&[Quotient[n, 2]];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Nov 18 2023, after Alois P. Heinz *)

a(n) = A003724(ceiling(n/2)) * A005046(floor(n/4)) if (n mod 4) in {0,1}.

a(n) = 0 if (n mod 4) in {2,3}.

A292966 a(n) = (2n)! [x^(2n)] exp(n(cosh(x)-1)).

Original entry on oeis.org

1, 1, 14, 543, 41332, 5203880, 979067634, 257327195587, 90055440034760, 40484356990454979, 22735298894247204310, 15597865046044378254146, 12836943134715746781979644, 12482280872844033169540407253, 14157380505669130674125989779482, 18524549200247005824873100782063015

Offset: 0

Ilya Gutkovskiy, Sep 29 2017

nonn

Cf. A005046, A293022.

Table[(2 n)! SeriesCoefficient[Exp[n (Cosh[x] - 1)], {x, 0, 2 n}], {n, 0, 15}]
Table[(-1)^n (2 n)! SeriesCoefficient[Exp[n (Cos[x] - 1)], {x, 0, 2 n}], {n, 0, 15}]

A302579 Expansion of e.g.f. exp(cosh(x)/cos(x)-1) (even powers only).

Original entry on oeis.org

1, 2, 24, 632, 28784, 1991552, 193410624, 24993180032, 4134783110144, 850499728758272, 212579274719007744, 63381008507902595072, 22200896917210834817024, 9019985888570141052280832, 4204783981520054371872374784, 2228007853953954434037178007552

Offset: 0

Ilya Gutkovskiy, Apr 10 2018

nonn

			exp(cosh(x)/cos(x)-1) = 1 + 2*x^2/2! + 24*x^4/4! + 632*x^6/6! + 28784*x^8/8! + ...

N. J. A. Sloane, Transforms

Cf. A000795, A005046, A006229, A009234, A009253, A009254, A009274, A012853, A217502.

nmax = 15; Table[(CoefficientList[Series[Exp[Cosh[x]/Cos[x] - 1], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]

a(n) = (2*n)! * [x^(2*n)] exp(cosh(x)/cos(x)-1).

A331818 E.g.f.: exp(1 - sec(x)) (even powers only).

Original entry on oeis.org

1, -1, -2, -1, 253, 12854, 668053, 39148199, 2456262898, 130790155859, -3853032641387, -4312625669814166, -1531200886955161127, -489884373969089299201, -159097972223555719000922, -54064488830901650420384521, -19284261543086608770504566147

Offset: 0

Ilya Gutkovskiy, Jan 27 2020

sign

Cf. A000364, A005046, A217502, A260884.

nmax = 16; Table[(CoefficientList[Series[Exp[1 - Sec[x]], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]
a[0] = 1; a[n_] := a[n] = -Sum[Binomial[2 n - 1, 2 k - 1] Abs[EulerE[2 k]] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 16}]

a(n) = {polcoef(serlaplace(exp(1 - 1/cos(x + O(x^(2*n + 1))))), 2*n)} \\ Andrew Howroyd, Jan 27 2020

a(0) = 1; a(n) = -Sum_{k=1..n} binomial(2*n-1,2*k-1) * A000364(k) * a(n-k).

A372508 Expansion of e.g.f. exp(sinh(x)) - exp(cosh(x) - 1).

Original entry on oeis.org

0, 1, 0, 2, 1, 12, 6, 128, 78, 1872, 1613, 37600, 38336, 990784, 1124280, 32333824, 41622181, 1272660224, 1843050734, 59527313920, 94591980910, 3252626013184, 5602035320753, 204354574172160, 380190518533920, 14594815769038848, 29179899891380592, 1174376539738169344

Offset: 0

Ilya Gutkovskiy, May 04 2024

nonn

Number of partitions of n-set into odd blocks minus number of partitions of n-set into even blocks.

Cf. A000110, A003724, A005046.

nmax = 27; CoefficientList[Series[Exp[Sinh[x]] - Exp[Cosh[x] - 1], {x, 0, nmax}], x] Range[0, nmax]!

cp's OEIS Frontend

A097322 (2n)! divided by denominator of Taylor expansion of exp(cos(x)-1).

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PARI

A327005 T(n, k) = Sum_{i=1..n} BM[k][i] where BM is the BellMatrix(x -> x mod n) as defined in A264428. Square array read by ascending antidiagonals for n >= 1 and k >= 1.

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Maple

A330041 Expansion of e.g.f. exp(cosh(exp(x) - 1) - 1).

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Maple

Mathematica

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A352145 Expansion of e.g.f. exp(-1 + cos(x) + sin(x)).

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PARI

PARI

Formula

A352465 a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} binomial(2*n,2*k)^2 * k * a(n-k).

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A363073 Number of set partitions of [n] such that each element is contained in a block whose block size parity coincides with the parity of the element.

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Maple

Mathematica

Formula

A292966 a(n) = (2*n)! * [x^(2*n)] exp(n*(cosh(x)-1)).

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Mathematica

A302579 Expansion of e.g.f. exp(cosh(x)/cos(x)-1) (even powers only).

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Mathematica

Formula

A331818 E.g.f.: exp(1 - sec(x)) (even powers only).

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A352465 a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} binomial(2n,2k)^2 * k * a(n-k).

A292966 a(n) = (2n)! [x^(2n)] exp(n(cosh(x)-1)).