A255884 -id:A255884 - cp's OEIS frontend

A255881 Expansion of exp( Sum_{n >= 1} A000364(n)*x^n/n ).

1, 1, 3, 23, 371, 10515, 461869, 28969177, 2454072147, 269732425859, 37312477130105, 6342352991066661, 1299300852841580893, 315702973949640373933, 89765549161833322593411, 29526682496433138896248775, 11124674379405792463701519059

Offset: 0

Peter Bala, Mar 09 2015

A000364(n) = (-1)^n*2^(2*n)*Euler(2*n,1/2), where E(n,x) is the n-th Euler polynomial. In general it appears that when k is a nonzero integer, the expansion of exp( Sum_{n >= 1} k^(2*n)*E(2*n,1/k)*(-x)^n/n ) has (positive) integer coefficients. See A255882 (k = 3), A255883(k = 4) and A255884 (k = 6).

G. C. Greubel, Table of n, a(n) for n = 0..200
E. W. Weisstein, Euler Polynomial

Cf. A000364, A188514, A255882, A255883, A255884.

#A255881
k := 2:
exp(add(k^(2*n)*euler(2*n, 1/k)*(-x)^n/n, n = 1 .. 16)): seq(coeftayl(%, x = 0, n), n = 0 .. 16);

A000364:= Table[Abs[EulerE[2 n]], {n, 0, 80}]; a:= With[{nmax = 70}, CoefficientList[Series[Exp[Sum[A000364[[k + 1]]*x^(k)/(k), {k, 1, 75}]], {x, 0, nmax}], x]]; Table[a[[n]], {n, 1, 50}] (* G. C. Greubel, Aug 26 2018 *)

O.g.f.: exp( x + 5*x^2/2 + 61*x^3/3 + 1385*x^4/4 + ... ) = 1 + x + 3*x^2 + 23*x^3 + 371*x^4 + ....
a(0) = 1 and for n >= 1, n*a(n) = Sum_{k = 1..n} (-1)^k*2^(2*k)*E(2*k,1/2)*a(n-k).
a(n) ~ 2^(4*n + 3) * n^(2*n - 1/2) / (Pi^(2*n + 1/2) * exp(2*n)). - Vaclav Kotesovec, Jun 08 2019

A255882 Expansion of exp( Sum_{n >= 1} A210657(n)*(-x)^n/n ).

Original entry on oeis.org

1, 2, 13, 224, 8170, 522716, 51749722, 7309866728, 1394040714169, 344865267322010, 107361980072755261, 41067497940750566312, 18931745446455458282248, 10350955324610065848650384, 6622526747212249020075069880, 4901565185965701578921602882976

Offset: 0

Peter Bala, Mar 09 2015

A210657(n) = 3^(2*n)*E(2*n,1/3), where E(n,x) is the n-th Euler polynomial. In general it appears that when is k a nonzero integer, the expansion of exp( Sum_{n >= 1} k^(2*n)*E(2*n,1/k)*(-x)^n/n ) has (positive) integer coefficients. See A255881 (k = 2), A255883(k = 4) and A255884 (k = 6).

G. C. Greubel, Table of n, a(n) for n = 0..200
E. W. Weisstein, Euler Polynomial

Cf. A188514, A210657, A255881, A255883, A255884.

#A255882
k := 3:
exp(add(k^(2*n)*euler(2*n, 1/k)*(-x)^n/n, n = 1 .. 15)): seq(coeftayl(%, x = 0, n), n = 0 .. 15);

A210657[n_]:= 9^n EulerE[2 n, 1/3]; a:= With[{nmax = 80}, CoefficientList[Series[Exp[Sum[A210657[k]*(-x)^(k)/(k), {k, 1, 75}]], {x, 0, nmax}], x]]; Table[a[[n]], {n, 1, 51}] (* G. C. Greubel, Aug 26 2018 *)

O.g.f.: exp( 2*x + 22*x^2/2 + 602*x^3/3 + 30742*x^4/4 + ... ) = 1 + 2*x + 13*x^2 + 224*x^3 + 8170*x^4 + ....
a(0) = 1 and for n >= 1, n*a(n) = Sum_{k = 1..n} (-1)^k*3^(2*k)*E(2*k,1/3)*a(n-k).
a(n) ~ 2^(2*n + 2) * 3^(2*n + 1/2) * n^(2*n - 1/2) / (exp(2*n) * Pi^(2*n + 1/2)). - Vaclav Kotesovec, Jun 08 2019

A002438 Multiples of Euler numbers.

Original entry on oeis.org

1, 5, 205, 22265, 4544185, 1491632525, 718181418565, 476768795646785, 417370516232719345, 465849831125196593045, 645702241048404020542525, 1088120580608731523115639305, 2190881346273790815462670984105

Offset: 1

N. J. A. Sloane

A. Fletcher, J. C. P. Miller, L. Rosenhead and L. J. Comrie, An Index of Mathematical Tables. Vols. 1 and 2, 2nd ed., Blackwell, Oxford and Addison-Wesley, Reading, MA, 1962, Vol. 1, p. 75.
Glaisher, J. W. L.; Messenger of Math., 28 (1898), 36-79, see esp. p. 51.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 1..100
P. Bala, A triangle for calculating A002438
P. Bala, Some S-fractions related to the expansions of sin(ax)/cos(bx) and cos(ax)/cos(bx)
Matthieu Josuat-Vergès and Jang Soo Kim, Touchard-Riordan formulas, T-fractions, and Jacobi's triple product identity, arXiv:1101.5608 [math.CO] (2011)

Cf. A000364, A086646, A255884.

a[n_] := (1+9^(n-1))*Abs[EulerE[2*(n-1)]]/2; Table[ a[n], {n, 1, 13}](* Jean-François Alcover, Feb 10 2012 *)

A002438(n)=A000364(n-1)*(9^(n-1)+1)\2 \\ - M. F. Hasler, Jul 21 2013

a(n) = A000364(n-1) * (9^(n-1) + 1)/2.
a(n+1) = Sum_{k = 0..n} A086646(n, k)*(-4)^k*9^(n-k). - Philippe Deléham, Aug 26 2005
From Peter Bala, Mar 13 2015: (Start)
a(n+1) = (-1)^n*6^(2*n)*E(2*n,1/6).
Assuming an offset of 0, the e.g.f. is cos(2*x)/cos(3*x) = 1 + 5*x + 205*x^2/2! + 22265*x^3/3! + 4544185*x^4/4! + ....
O.g.f. as a continued fraction: x/(1 - (3^2 - 2^2)*x/(1 - 6^2*x/(1 - (9^2 - 2^2)*x/(1 - 12^2*x/(1 - ... ))))) = x + 5*x^2 + 205*x^3 + 22265*x^4 + 4544185*x^5 + .... See Josuat-Vergès and Kim, p. 23. Cf. A086646.
The expansion of exp( Sum_{n >= 1} a(n+1)*x^n/n ) = exp( 5*x + 205*x^2/2 + 22265*x^3/3 + 4544185 *x^4/4 + ... ) appears to have integer coefficients. See A255884.
(End)
From Peter Bala, Nov 10 2015: (Start)
O.g.f. A(x) = 1/(1 + x - 6*x/(1 - 30*x/(1 + x - 84*x/(1 - 132*x/(1 + x - ... - 6*n*(6*n - 5)*x/(1 - 6*n*(6*n - 1)*x/(1 + x - ))))))).
A(x) = 1/(1 + 25*x - 30*x/(1 - 6*x/(1 + 25*x - 132*x/(1 - 84*x/(1 + 25*x - ... - 6*n*(6*n - 1)*x/(1 - 6*n*(6*n - 5)*x/(1 + 25*x - ))))))). (End)

More terms from Herman P. Robinson

More terms from Jon E. Schoenfield, May 09 2010

A255883 Expansion of exp( Sum_{n >= 1} A000281(n)*x^n/n ).

Original entry on oeis.org

1, 3, 33, 1011, 65985, 7536099, 1329205857, 334169853267, 113370124235649, 49880529542872515, 27614111852126579361, 18782012442066306225843, 15394836674855296870428993, 14965462261283347594195897251, 17023467576167762236198869304545, 22400927665017118737825435362462739

Offset: 0

Peter Bala, Mar 09 2015

A000281(n) =(-1)^n*4^(2*n)*E(2*n,1/4), where E(n,x) denotes the n-th Euler polynomial. In general it appears that when k is a nonzero integer, the expansion of exp( Sum_{n >= 1} k^(2*n)*E(2*n,1/k)*(-x)^n/n ) has (positive) integer coefficients. See A255881 (k = 2), A255882(k = 3) and A255884 (k = 6).

G. C. Greubel, Table of n, a(n) for n = 0..200
E. W. Weisstein, Euler Polynomial

Cf. A000281, A188514, A255881, A255882, A255884

#A255883
k := 4:
exp(add(k^(2*n)*euler(2*n, 1/k)*(-x)^n/n, n = 1 .. 15)): seq(coeftayl(%, x = 0, n), n = 0 .. 15);

A000281:= With[{nn = 200}, Take[CoefficientList[Series[Cos[x]/Cos[2 x], {x, 0, nn}], x] Range[0, nn]!, {1, -1, 2}]]; a:= With[{nmax = 80}, CoefficientList[Series[Exp[Sum[A000281[[k + 1]]*x^(k)/(k), {k, 1, 85}]], {x, 0, nmax}], x]]; Table[a[[n]], {n, 1, 50}]  (* G. C. Greubel, Aug 26 2018 *)

O.g.f.: exp( 3*x + 57*x^2/2 + 2763*x^3/3 + 250737*x^4/4 + ... ) = 1 + 3*x + 33*x^2 + 1011*x^3 + 65985*x^4 + ....
a(0) = 1 and for n >= 1, n*a(n) = Sum_{k = 1..n} (-1)^k*4^(2*k)*E(2*k,1/4)*a(n-k).
a(n) ~ 2^(6*n + 5/2) * n^(2*n - 1/2) / (exp(2*n) * Pi^(2*n + 1/2)). - Vaclav Kotesovec, Jun 08 2019

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A255881 Expansion of exp( Sum_{n >= 1} A000364(n)*x^n/n ).

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A255882 Expansion of exp( Sum_{n >= 1} A210657(n)*(-x)^n/n ).

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A002438 Multiples of Euler numbers.

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A255883 Expansion of exp( Sum_{n >= 1} A000281(n)*x^n/n ).

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