A065143 -id:A065143 - cp's OEIS frontend

A199572 Number of round trips of length n on the cycle graph C_2 from any of the two vertices.

1, 0, 4, 0, 16, 0, 64, 0, 256, 0, 1024, 0, 4096, 0, 16384, 0, 65536, 0, 262144, 0, 1048576, 0, 4194304, 0, 16777216, 0, 67108864, 0, 268435456, 0, 1073741824, 0, 4294967296, 0, 17179869184, 0, 68719476736, 0, 274877906944, 0

Offset: 0

Wolfdieter Lang, Nov 08 2011

See the array and triangle A199571 for the general cycle graph C_N counting.
This is A000302 and A000004 interleaved. - Omar E. Pol, Nov 09 2011
With offset = 1:  Number of ways to separate n distinguishable objects into an odd size pile and an even size pile.  For example:  a(3) = 4 because we have: {{1},{2,3}}; {{2},{1,3}}; {{3},{1,2}}; {{1,2,3},{}}. - Geoffrey Critzer, Jun 10 2013
Inverse Stirling transform of A065143. - Vladimir Reshetnikov, Nov 01 2015

			a(2) = 4 from starting with vertex no. 1, with edges e1 and e2 to vertex no. 2: e1e1, e2e2, e1e2 and e2e1.

R. J. Mathar, Counting Walks on Finite Graphs, Section 1.
Index entries for linear recurrences with constant coefficients, signature (0,4).

Cf. A000007 (N=1), A078008 (N=3). a(n) is second row of array w(N,L) A199571, and second column of the triangle a(K,N) A199571.

Cf. A065143 (Stirling transform).

nn = 39; Drop[Range[0, nn]! CoefficientList[Series[ Sinh[x] Cosh[x], {x, 0, nn}],x], 1] (* Geoffrey Critzer, Jun 10 2013 *)

vector(100, n, n--; (2^(n) +(-2)^n)/2) \\ Altug Alkan, Nov 02 2015

a(n) = (2^n + (-2)^n)/2 = 2^(n-1)*(1 + (-1)^n).
O.g.f.: 1/(1-(2*x)^2).
E.g.f.: cosh(2*x)=U(0) where U(k) = 1 + 2*x^2/((4*k+1)*(2*k+1) - x^2*(4*k+1)*(2*k+1)/(x^2 + (4*k+3)*(k+1)/U(k+1))); (continued fraction). - Sergei N. Gladkovskii, Oct 23 2012

A264036 Stirling transform of A077957 (aerated powers of 2).

Original entry on oeis.org

1, 0, 2, 6, 18, 70, 330, 1694, 9202, 53334, 332090, 2212782, 15638370, 116365990, 907975146, 7413080510, 63212284498, 561747543414, 5190343710746, 49752410984526, 493844719701186, 5068209425457862, 53705511911500746, 586862875255860062, 6605213319604075186

Offset: 0

Vladimir Reshetnikov, Nov 01 2015

nonn

a(n) is the inverse binomial transform of A264037 without the leading zero [1, 1, 3, 13, 55, ...].

			G.f. = 1 + 2*x^2 + 6*x^3 + 18*x^4 + 70*x^5 + 330*x^6 + 1694*x^7 + 9202*x^8 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..567
Eric Weisstein's MathWorld, Bell Polynomial.

Column k=2 of A357681.

Cf. A065143, A077957, A264037.

Table[(BellB[n, Sqrt[2]] + BellB[n, -Sqrt[2]])/2, {n, 0, 24}]

vector(100, n, n--; sum(k=0, n\2, 2^k*stirling(n, 2*k, 2))) \\ Altug Alkan, Nov 01 2015

a(n) = Sum_{k=0..n} A077957(k)*Stirling2(n,k).
a(n) = Sum_{k=0..floor(n/2)} 2^k*Stirling2(n,2*k).
a(n) = (Bell_n(sqrt(2)) + Bell_n(-sqrt(2)))/2, where Bell_n(x) is n-th Bell polynomial.
Bell_n(sqrt(2)) = a(n) + A264037(n)*sqrt(2).
E.g.f.: cosh(sqrt(2)*(exp(x) - 1)).
a(n) = 1; a(n) = 2 * Sum_{k=0..n-1} binomial(n-1, k) * A264037(k). - Seiichi Manyama, Oct 12 2022

A357598 Expansion of e.g.f. sinh(2 * (exp(x)-1)) / 2.

Original entry on oeis.org

0, 1, 1, 5, 25, 117, 601, 3509, 22457, 153141, 1105561, 8453557, 68339833, 581495605, 5184047961, 48259748533, 468040609593, 4719817792565, 49396003390489, 535526127566773, 6004124908829177, 69509047405180213, 829801009239621849, 10202835010223731893

Offset: 0

Seiichi Manyama, Oct 05 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..558
Eric Weisstein's World of Mathematics, Bell Polynomial.

Cf. A024429, A264037, A357572.

Cf. A065143, A078944, A357599.

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

a(n) = sum(k=0, (n-1)\2, 4^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, 2)-Bell_poly(n, -2)))/4;

a(n) = Sum_{k=0..floor((n-1)/2)} 4^k * Stirling2(n,2*k+1).
a(n) = ( Bell_n(2) - Bell_n(-2) )/4, where Bell_n(x) is n-th Bell polynomial.
a(n) = 0; a(n) = Sum_{k=0..n-1} binomial(n-1, k) * A065143(k).

A357681 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. cosh( sqrt(k) * (exp(x) - 1) ).

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 2, 3, 0, 1, 0, 3, 6, 8, 0, 1, 0, 4, 9, 18, 25, 0, 1, 0, 5, 12, 30, 70, 97, 0, 1, 0, 6, 15, 44, 135, 330, 434, 0, 1, 0, 7, 18, 60, 220, 705, 1694, 2095, 0, 1, 0, 8, 21, 78, 325, 1228, 3906, 9202, 10707, 0, 1, 0, 9, 24, 98, 450, 1905, 7196, 22953, 53334, 58194, 0

Offset: 0

Seiichi Manyama, Oct 09 2022

			Square array begins:
  1,  1,  1,   1,   1,   1, ...
  0,  0,  0,   0,   0,   0, ...
  0,  1,  2,   3,   4,   5, ...
  0,  3,  6,   9,  12,  15, ...
  0,  8, 18,  30,  44,  60, ...
  0, 25, 70, 135, 220, 325, ...

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

Columns k=0-4 give: A000007, A024430, A264036, A357615, A065143.
Column k=9 gives A357667.
Main diagonal gives A357682.
Cf. A292860.

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

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

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

T(n,k) = ( Bell_n(sqrt(k)) + Bell_n(-sqrt(k)) )/2, where Bell_n(x) is n-th Bell polynomial.

A357615 Expansion of e.g.f. cosh(sqrt(3) * (exp(x) - 1)).

Original entry on oeis.org

1, 0, 3, 9, 30, 135, 705, 3906, 22953, 145053, 985800, 7136613, 54544485, 437961888, 3685605735, 32441696325, 297977767662, 2848636972971, 28278241848309, 290931124989546, 3097051613077269, 34064462020306473, 386600759467746528, 4521440483724439521

Offset: 0

Seiichi Manyama, Oct 06 2022

nonn

			G.f. = 1 + 3*x^2 + 9*x^3 + 30*x^4 + 135*x^5 + 705*x^6 + ... - _Michael Somos_, Oct 06 2022

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

Column k=3 of A357681.
Cf. A065143, A264036.
Cf. A357572.

a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ Cosh[Sqrt[3] * (Exp@x - 1)], {x, 0, n}]]; (* Michael Somos, Oct 06 2022 *)

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

my(x='x+O('x^30)); apply(round, Vec(serlaplace(cosh(sqrt(3) * (exp(x) - 1))))) \\ Michel Marcus, Oct 06 2022

{a(n) = if(n<0, 0, n!*simplify(polcoeff( cosh(quadgen(12) * (exp(x + x*O(x^n)) - 1)), n)))}; /* Michael Somos, Oct 06 2022 */

a(n) = Sum_{k=0..floor(n/2)} 3^k * Stirling2(n,2*k).
a(n) = ( Bell_n(sqrt(3)) + Bell_n(-sqrt(3)) )/2, where Bell_n(x) is n-th Bell polynomial.
a(n) = 1; a(n) = 3 * Sum_{k=0..n-1} binomial(n-1, k) * A357572(k).

A009599 Expansion of e.g.f. sinh(sinh(x)*exp(x)).

Original entry on oeis.org

0, 1, 2, 5, 20, 117, 782, 5441, 39496, 306921, 2602682, 24116413, 241121564, 2561633245, 28613237382, 334511450617, 4089814554384, 52302564139985, 699179303859698, 9751200460426357, 141494250613386916

Offset: 0

R. H. Hardin

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

Cf. A065143.

[(&+[2^(n-k)*StirlingSecond(n,k)*(1 - (-1)^k)/2: k in [0..n]]): n in [0..30]]; // G. C. Greubel, Jan 22 2018

Table[Sum[StirlingS2[n, k]*(1-(-1)^k)/2*2^(n-k),{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Aug 06 2014 after Vladeta Jovovic *)
Table[(BellB[n, 1/2] - BellB[n, -1/2]) 2^(n-1), {n, 0, 20}] (* Vladimir Reshetnikov, Nov 01 2015 *)
With[{nn=20},CoefficientList[Series[Sinh[Sinh[x]Exp[x]],{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, Jun 02 2017 *)

x='x+O('x^30); concat([0], Vec(serlaplace(sinh(sinh(x)*exp(x))))) \\ G. C. Greubel, Jan 22 2018

a(n) = Sum_{k=0..n} Stirling2(n, k)*(1-(-1)^k)/2*2^(n-k). - Vladeta Jovovic, Sep 26 2003
G.f.: Sum_{k>=0} x^(2*k+1)/Product_{i=0..2*k+1} (1 - 2*i*x). - Sergei N. Gladkovskii, Jan 06 2013
G.f.: x/( G(0)-x^2 ) where G(k) = x^2 + (4*x*k-1)*(4*x*k+2*x-1) - x^2*(4*x*k-1)*(4*x*k+2*x-1)/G(k+1); (recursively defined continued fraction). - Sergei N. Gladkovskii, Jan 06 2013

Extended and signs tested by Olivier Gérard, Mar 15 1997

A009153 Expansion of e.g.f. cosh(sinh(x)*exp(x)).

Original entry on oeis.org

1, 0, 1, 6, 29, 140, 757, 4858, 36409, 302520, 2681769, 25018510, 245905365, 2559272196, 28264854685, 330408571202, 4065526003313, 52349977261040, 702393407898705, 9795673312888214, 141820637175889805

Offset: 0

R. H. Hardin

Vincenzo Librandi, Table of n, a(n) for n = 0..100
Vaclav Kotesovec, Asymptotic solution of the equations using the Lambert W-function

Cf. A009599, A065143.

CoefficientList[Series[Cosh[Sinh[x]*E^x], {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Aug 06 2014 *)
Table[Sum[StirlingS2[n, k]*(1+(-1)^k)/2*2^(n-k),{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Aug 06 2014 after Vladeta Jovovic *)
Table[(BellB[n, 1/2] + BellB[n, -1/2]) 2^(n-1), {n, 0, 20}] (* Vladimir Reshetnikov, Nov 01 2015 *)

a(n) = sum(k=0, n, stirling(n, k, 2)*(1+(-1)^k)/2*2^(n-k)); \\ Michel Marcus, Nov 02 2015

a(n) = Sum_{k=0..n} Stirling2(n, k)*(1+(-1)^k)/2*2^(n-k). - Vladeta Jovovic, Sep 26 2003
G.f.: 1 + Sum_{k>=0} x^(2*k+2)/Product_{i=0..2*k+2} (1-2*i*x). - Sergei N. Gladkovskii, Jan 06 2013
G.f.: 1 + x^2/( G(0)-x^2 ) where G(k) = x^2 + (4*x*k+2*x-1)*(4*x*k+4*x-1) - x^2*(4*x*k+2*x-1)*(4*x*k+4*x-1)/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Jan 06 2013
a(n) ~ cosh(exp(r)*sinh(r)) * n^(n+1/2) / (r^(n+1/2) * exp(n+r) * sqrt(exp(2*r) * r * sech(exp(r)*sinh(r))^2 + (1+2*r) * tanh(exp(r)*sinh(r)))), where r is the root of the equation r*exp(r)*(cosh(r) + sinh(r))*tanh(exp(r)*sinh(r)) = n. - Vaclav Kotesovec, Aug 06 2014

Extended and signs tested by Olivier Gérard, Mar 15 1997

A357727 Expansion of e.g.f. cos( 2 * (exp(x) - 1) ).

Original entry on oeis.org

1, 0, -4, -12, -12, 100, 852, 4004, 9940, -36828, -726316, -6174300, -35968812, -109708508, 702818004, 16677814436, 188794428628, 1542659688996, 8359981681364, -3068614764636, -868989327994668, -15076627082974940, -179727483880747308

Offset: 0

Seiichi Manyama, Oct 10 2022

sign

Andrew Howroyd, Table of n, a(n) for n = 0..200
Eric Weisstein's World of Mathematics, Bell Polynomial.

Column k=4 of A357728.

Cf. A065143, A357719, A357738.

my(N=30, x='x+O('x^N)); Vec(serlaplace(cos(2*(exp(x)-1))))

a(n) = sum(k=0, n\2, (-4)^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, 2*I)+Bell_poly(n, -2*I)))/2;

a(n) = Sum_{k=0..floor(n/2)} (-4)^k * Stirling2(n,2*k).
a(n) = 1; a(n) = -4 * Sum_{k=0..n-1} binomial(n-1, k) * A357738(k).
a(n) = ( Bell_n(2 * i) + Bell_n(-2 * i) )/2, where Bell_n(x) is n-th Bell polynomial and i is the imaginary unit.

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).

cp's OEIS Frontend

A199572 Number of round trips of length n on the cycle graph C_2 from any of the two vertices.

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A264036 Stirling transform of A077957 (aerated powers of 2).

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A357598 Expansion of e.g.f. sinh(2 * (exp(x)-1)) / 2.

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A357681 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. cosh( sqrt(k) * (exp(x) - 1) ).

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A357615 Expansion of e.g.f. cosh(sqrt(3) * (exp(x) - 1)).

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A009599 Expansion of e.g.f. sinh(sinh(x)*exp(x)).

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A009153 Expansion of e.g.f. cosh(sinh(x)*exp(x)).

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