cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 21-30 of 30 results.

A352251 Expansion of e.g.f. 1 / (1 - x * sinh(x)) (even powers only).

Original entry on oeis.org

1, 2, 28, 966, 62280, 6452650, 980531916, 205438870014, 56760128400016, 19994672935658322, 8746764024725937300, 4651991306703670964518, 2956156902003429777549144, 2212026607642404922284728826, 1925137044528752884360406444380, 1928103808741894922401976601295950
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 09 2022

Keywords

Crossrefs

Cf. A006153, A006154, A009233, A094088, A205571, A210672, A351762, A352250, A352252, A352253.

Programs

  • Mathematica
    nmax = 30; Take[CoefficientList[Series[1/(1 - x Sinh[x]), {x, 0, nmax}], x] Range[0, nmax]!, {1, -1, 2}]
a[0] = 1; a[n_] := a[n] = 2 Sum[Binomial[2 n, 2 k] k a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 15}]
  • PARI
    my(x='x+O('x^40), v=Vec(serlaplace(1 /(1-x*sinh(x))))); vector(#v\2, k, v[2*k-1]) \\ Michel Marcus, Mar 10 2022

Formula

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

A352326 Expansion of e.g.f.: 1/(2 - exp(x) - sinh(x)).

Original entry on oeis.org

1, 2, 9, 62, 567, 6482, 88929, 1423382, 26037027, 535813802, 12251630349, 308153112302, 8455276083087, 251333936555522, 8045613346221369, 275950004166050822, 10095559110771678747, 392427366313299119642, 16151459739717643489989
Offset: 0

Views

Author

Seiichi Manyama, Mar 12 2022

Keywords

Links

Crossrefs

Cf. A000670, A006154, A028296, A094088, A352327, A352617.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=0, 1, add(
      a(n-k)*binomial(n, k)*(1+(k mod 2)), k=1..n))
    end:
seq(a(n), n=0..18); # Alois P. Heinz, Mar 25 2022
  • Mathematica
    m = 18; Range[0, m]! * CoefficientList[Series[1/(2 - Exp[x] - Sinh[x]), {x, 0, m}], x] (* Amiram Eldar, Mar 12 2022 *)
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(2-exp(x)-sinh(x))))
  • PARI
    a(n) = if(n==0, 1, sum(k=1, n, (3-(-1)^k)/2*binomial(n, k)*a(n-k)));

Formula

a(0) = 1; a(n) = Sum_{k=1..n} (3-(-1)^k)/2 * binomial(n,k) * a(n-k).
a(n) ~ n! / (sqrt(7) * log((2 + sqrt(7))/3)^(n+1)). - Vaclav Kotesovec, Mar 12 2022

A352327 Expansion of e.g.f.: 1/(3 - exp(x) - cosh(x)).

Original entry on oeis.org

1, 1, 4, 19, 130, 1081, 10894, 127639, 1711210, 25798141, 432212134, 7964801659, 160121522290, 3487254825601, 81790592435374, 2055350489070079, 55093108433421370, 1569052795651631461, 47315282424232826614, 1506074331671551028899
Offset: 0

Views

Author

Seiichi Manyama, Mar 12 2022

Keywords

Links

Crossrefs

Cf. A000670, A006154, A094088, A352326, A352624.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=0, 1, add(
      a(n-k)*binomial(n, k)*(2-(k mod 2)), k=1..n))
    end:
seq(a(n), n=0..19); # Alois P. Heinz, Mar 25 2022
  • Mathematica
    m = 19; Range[0, m]! * CoefficientList[Series[1/(3 - Exp[x] - Cosh[x]), {x, 0, m}], x] (* Amiram Eldar, Mar 12 2022 *)
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(3-exp(x)-cosh(x))))
  • PARI
    a(n) = if(n==0, 1, sum(k=1, n, (3+(-1)^k)/2*binomial(n, k)*a(n-k)));

Formula

a(0) = 1; a(n) = Sum_{k=1..n} (3+(-1)^k)/2 * binomial(n,k) * a(n-k).
a(n) ~ n! / (sqrt(6) * log(1 + sqrt(2/3))^(n+1)). - Vaclav Kotesovec, Mar 12 2022

A352253 Expansion of e.g.f. 1 / (1 - x * sinh(x) / 2) (even powers only).

Original entry on oeis.org

1, 1, 8, 153, 5492, 316625, 26774622, 3121729709, 479962730648, 94087054172673, 22904161764512570, 6778870099212235805, 2397161662661680925364, 998186321121004312238513, 483430830256916593106991782, 269435322393253822641626419725, 171224984800186115316322226731952
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 09 2022

Keywords

Crossrefs

Cf. A006153, A006154, A094088, A205571, A352251, A352254.

Programs

  • Mathematica
    nmax = 32; Take[CoefficientList[Series[1/(1 - x Sinh[x]/2), {x, 0, nmax}], x] Range[0, nmax]!, {1, -1, 2}]
a[0] = 1; a[n_] := a[n] = Sum[Binomial[2 n, 2 k] k a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 16}]
  • PARI
    my(x='x+O('x^40), v=Vec(serlaplace(1 /(1-x*sinh(x)/2)))); vector(#v\2, k, v[2*k-1]) \\ Michel Marcus, Mar 10 2022

Formula

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

A352467 a(0) = 1; a(n) = Sum_{k=1..n} binomial(2*n,2*k)^2 * a(n-k).

Original entry on oeis.org

1, 1, 37, 8551, 6886069, 14323022551, 64085654997739, 545107167737695109, 8062740187879748199029, 193866963305030079530064391, 7188682292472952994057436691387, 394013888612808806428687953794890229, 30829606055995735731623164115609901072859
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 17 2022

Keywords

Crossrefs

Cf. A094088, A102221, A352468, A352470.

Programs

  • Mathematica
    a[0] = 1; a[n_] := a[n] = Sum[Binomial[2 n, 2 k]^2 a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 12}]
nmax = 24; Take[CoefficientList[Series[1/(1 - Sum[x^(2 k)/(2 k)!^2, {k, 1, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!^2, {1, -1, 2}]

Formula

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

A352468 a(0) = 1; a(n) = Sum_{k=1..n} binomial(2*n,2*k)^3 * a(n-k).

Original entry on oeis.org

1, 1, 217, 735751, 16225658905, 1485378967457251, 429009059656530602767, 324779065084721999818137709, 563805297587600177760431368896025, 2028620600892240327820781003315525267467, 13978450121866685445815888094629703793828769467
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 17 2022

Keywords

Crossrefs

Cf. A094088, A336195, A352467, A352471.

Programs

  • Mathematica
    a[0] = 1; a[n_] := a[n] = Sum[Binomial[2 n, 2 k]^3 a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 10}]
nmax = 20; Take[CoefficientList[Series[1/(1 - Sum[x^(2 k)/(2 k)!^3, {k, 1, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!^3, {1,-1, 2}]

Formula

Sum_{n>=0} a(n) * x^(2*n) / (2*n)!^3 = 1 / (1 - Sum_{n>=1} x^(2*n) / (2*n)!^3).

A348587 Expansion of e.g.f. exp(x) / (2 - cos(x)).

Original entry on oeis.org

1, 1, 0, -2, 2, 26, -30, -622, 982, 25846, -50910, -1639142, 3874862, 147434366, -406614390, -17851478062, 56266545142, 2799621404086, -9927225631470, -552054087163382, 2175042302117822, 133686372253841006, -579383205000618150, -39002628245713951102
Offset: 0

Views

Author

Ilya Gutkovskiy, Oct 24 2021

Keywords

Crossrefs

Cf. A003701, A094088, A327034, A348580.

Programs

  • Mathematica
    nmax = 23; CoefficientList[Series[Exp[x]/(2 - Cos[x]), {x, 0, nmax}], x] Range[0, nmax]!
  • PARI
    my(x='x+O('x^40)); Vec(serlaplace(exp(x)/(2-cos(x)))) \\ Michel Marcus, Oct 24 2021

Formula

a(n) = Sum_{k=0..floor(n/2)} (-1)^k * binomial(n,2*k) * A094088(k).

A214407 Triangle read by rows. The coefficients in ascending order of x^i of the polynomials p{0}(x) = 1 and p{n}(x) = Sum_{k=0..n-1; k even} binomial(n, k) * p{k}(0)*(1 - n%2 + x^(n - k)).

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 0, 3, 0, 1, 7, 0, 6, 0, 1, 0, 35, 0, 10, 0, 1, 121, 0, 105, 0, 15, 0, 1, 0, 847, 0, 245, 0, 21, 0, 1, 3907, 0, 3388, 0, 490, 0, 28, 0, 1, 0, 35163, 0, 10164, 0, 882, 0, 36, 0, 1, 202741, 0, 175815, 0, 25410, 0, 1470, 0, 45, 0, 1, 0
Offset: 0

Views

Author

Peter Luschny, Jul 16 2012

Keywords

Comments

Matrix inverse of a signed variant of A119467.

Examples

			1
0, 1
1, 0, 1
0, 3, 0, 1
7, 0, 6, 0, 1
0, 35, 0, 10, 0, 1
121, 0, 105, 0, 15, 0, 1
0, 847, 0, 245, 0, 21, 0, 1
3907, 0, 3388, 0, 490, 0, 28, 0, 1

Crossrefs

Cf. A119467, A327034 (row sums), A094088 (column 0).

Programs

  • Sage
    @CachedFunction
def A214407_poly(n, x) :
    return 1 if n==0 else add(A214407_poly(k, 0)*binomial(n, k)*(x^(n-k)+1-n%2) for k in range(n)[::2])
def A214407_row(n) :
    R = PolynomialRing(ZZ, 'x')
    return R(A214407_poly(n,x)).coeffs()
for n in (0..8) : A214407_row(n)

Formula

T(n, k) = n! * [y^k] [x^n] (exp(x * y) / (2 - cosh(x))). - Peter Luschny, May 06 2023

A356900 a(n) = P(n, 1/2) where P(n, x) = x^(-n)*Sum_{k=0..n} A241171(n, k)*x^k.

Original entry on oeis.org

1, 1, 8, 154, 5552, 321616, 27325088, 3200979664, 494474723072, 97390246272256, 23820397371219968, 7083386168647642624, 2516691244849530785792, 1052914814802404260765696, 512347915163742179541659648, 286902390859642414913802102784, 183187476890368376930869730803712
Offset: 0

Views

Author

Peter Luschny, Sep 03 2022

Keywords

Comments

Other special values of this Euler type polynomials are: P(n, -1) = A000364(n); P(n, -1/2) = A002105(n); P(n, 1) = A094088(n), where we always make the assumption that the offset of the sequences is 0. A partition refinement of Joffe's triangle A241171 is A327022.

Crossrefs

Cf. A241171, A327022, A000364, A002105, A094088, A269941.

Programs

  • Maple
    a := n -> 2^n*add(A241171(n, k)*(1/2)^k, k = 0..n):
seq(a(n), n = 0..16);
  • SageMath
    # Using function PtransMatrix from A269941.
def E(n, v):
    eulr = lambda n: 1 / ((2 * n - 1) * (2 * n))
    norm = lambda n, k: (1 / v)^n * factorial(2 * n)
    P = PtransMatrix(n, eulr, norm)
    return [(-1)^j * sum([v^k * P[j][k] for k in range(j + 1)]) for j in range(n)]
A356900List = lambda n: E(n, -1/2); print(A356900List(17))
# A002105List = lambda n: E(n, 1/2) returns the reduced tangent numbers A002105.

A375381 E.g.f.: (exp(-x) - exp(x) - 2)/(exp(-x) + exp(x) - 4).

Original entry on oeis.org

1, 1, 1, 4, 7, 46, 121, 1114, 3907, 46246, 202741, 2933074, 15430207, 263817646, 1619195761, 31943268634, 224061282907, 5009616448246, 39531606447181, 987840438629794, 8661323866026007, 239217148602642046, 2307185279184885001, 69790939492563608554
Offset: 0

Views

Author

Peter Luschny, Aug 25 2024

Keywords

Crossrefs

Cf. A094088 (even bisection), A331978 (odd bisection).

Programs

  • Maple
    a := proc(n) option remember; local j;
ifelse(n = 0, 1, add(binomial(n, j) * a(j), j = 0..n-1, 2)) end:
# Or:
gf := (exp(-x) - exp(x) - 2)/(exp(-x) + exp(x) - 4):
series(gf, x, 24): seq(n!*coeff(%, x, n), n = 0..23);

Formula

a(n) = n! * [x^n] (1 + sinh(x))/(2 - cosh(x)).
a(n) = Sum_{j=0..n-1,2} binomial(n, j) * a(j) for n > 0, a(0) = 1. (Note that the sum runs in steps of 2.)
a(n) ~ n! * (1 + 1/sqrt(3) + (-1)^n * (-1 + 1/sqrt(3))) / log(2 + sqrt(3))^(n+1). - Vaclav Kotesovec, Sep 02 2024
Previous Showing 21-30 of 30 results.

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