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.

Showing 1-6 of 6 results.

A097397 Coefficients in asymptotic expansion of normal probability function.

Original entry on oeis.org

1, 1, 1, 5, 9, 129, 57, 9141, -36879, 1430049, -15439407, 418019205, -7404957255, 196896257505, -4656470025015, 134136890777205, -3845524501226655, 123250625100419265, -4085349586734306015, 145973136800663973765
Offset: 0

Views

Author

Michael Somos, Aug 13 2004

Keywords

Comments

a(0) + a(1)*x/(1-2*x) + a(2)*x^2/((1-2*x)*(1-4*x)) + ... = 1 + x + 3*x^2 + 15*x^3 + ...

References

  • M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 932.

Links

Crossrefs

Cf. A006252, A352070, A352073.

Programs

  • Mathematica
    Table[Sum[2^(n - 2*k)*(2*k)!/k! * SeriesCoefficient[(1 - n + x)*Pochhammer[2 - n + x, -1 + n], {x, 0, k}], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Aug 10 2019 *)
  • PARI
    a(n)=sum(k=0,n, 2^(n-2*k)*(2*k)!/k!* polcoeff(prod(i=0,n-1,x-i),k))
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(1/sqrt(1-log(1+2*x)))) \\ Seiichi Manyama, Mar 05 2022

Formula

E.g.f.: 1/sqrt(1 - log(1 + 2*x)). - Seiichi Manyama, Mar 05 2022
a(n) ~ n! * (-1)^(n+1) * 2^(n-1) / (log(n)^(3/2) * n) * (1 - 3*(gamma + 1)/(2*log(n)) + 15*(1 + 2*gamma + gamma^2 - Pi^2/6) / (8*log(n)^2)), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Mar 05 2022
From Seiichi Manyama, Nov 18 2023: (Start)
a(n) = Sum_{k=0..n} 2^(n-k) * (Product_{j=0..k-1} (2*j+1)) * Stirling1(n,k).
a(0) = 1; a(n) = Sum_{k=1..n} (-2)^k * (1/2 * k/n - 1) * (k-1)! * binomial(n,k) * a(n-k). (End)

A352119 Expansion of e.g.f. 1/(2 - exp(4*x))^(1/4).

Original entry on oeis.org

1, 1, 9, 121, 2289, 56401, 1713849, 61939081, 2595199329, 123690992161, 6608289658089, 391154820258841, 25408740616159569, 1797051730819428721, 137463201511019813529, 11308020549364112399401, 995455518982520306979009, 93373681491447943767190081
Offset: 0

Views

Author

Seiichi Manyama, Mar 05 2022

Keywords

Crossrefs

Cf. A000670, A352117, A352118.
Cf. A352073.

Programs

  • Mathematica
    m = 17; Range[0, m]! * CoefficientList[Series[(2 - Exp[4*x])^(-1/4), {x, 0, m}], x] (* Amiram Eldar, Mar 05 2022 *)
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(2-exp(4*x))^(1/4)))
  • PARI
    a(n) = sum(k=0, n, 4^(n-k)*prod(j=0, k-1, 4*j+1)*stirling(n, k, 2));

Formula

a(n) = Sum_{k=0..n} 4^(n-k) * (Product_{j=0..k-1} (4*j+1)) * Stirling2(n,k).
a(n) ~ n! * 2^(2*n - 1/4) / (Gamma(1/4) * n^(3/4) * log(2)^(n + 1/4)). - Vaclav Kotesovec, Mar 05 2022
From Seiichi Manyama, Nov 18 2023: (Start)
a(0) = 1; a(n) = Sum_{k=1..n} 4^k * (1 - 3/4 * k/n) * binomial(n,k) * a(n-k).
a(0) = 1; a(n) = a(n-1) - 2*Sum_{k=1..n-1} (-4)^k * binomial(n-1,k) * a(n-k). (End)

A352070 Expansion of e.g.f. 1/(1 - log(1 + 3*x))^(1/3).

Original entry on oeis.org

1, 1, 1, 10, 10, 604, -1844, 107344, -1201400, 42193576, -875584376, 29853569008, -880141783184, 32865860907424, -1216481572723616, 51296026356128512, -2244334822166729600, 106984479644794783360, -5358207684820194270080, 286466413246622566048000
Offset: 0

Views

Author

Seiichi Manyama, Mar 05 2022

Keywords

Crossrefs

Cf. A006252, A097397, A352073.
Cf. A007559, A133480, A352113, A352118.

Programs

  • Mathematica
    m = 19; Range[0, m]! * CoefficientList[Series[(1 - Log[1 + 3*x])^(-1/3), {x, 0, m}], x] (* Amiram Eldar, Mar 05 2022 *)
Table[Sum[3^(n-k) * Product[3*j+1, {j,0,k-1}] * StirlingS1[n,k], {k,0,n}], {n,0,20}] (* Vaclav Kotesovec, Sep 07 2023 *)
  • Maxima
    a[n]:=if n=0 then 1 else n!*sum(a[n-k]*(2/n/3-1/k)*(-3)^k/(n-k)!,k,1,n);
makelist(a[n],n,0,50); /* Tani Akinari, Sep 07 2023 */
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1-log(1+3*x))^(1/3)))
  • PARI
    a(n) = sum(k=0, n, 3^(n-k)*prod(j=0, k-1, 3*j+1)*stirling(n, k, 1));

Formula

a(n) = Sum_{k=0..n} 3^(n-k) * (Product_{j=0..k-1} (3*j+1)) * Stirling1(n,k).
For n > 0, a(n) = n!*Sum_{k=1..n} a(n-k)*(2/n/3-1/k)*(-3)^k/(n-k)!. - Tani Akinari, Sep 07 2023
a(n) ~ -(-1)^n * 3^(n-1) * n! / (n * log(n)^(4/3)) * (1 - 4*(1+gamma)/(3*log(n))), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Sep 07 2023

A352114 Expansion of e.g.f. (1 - log(1 - 4*x))^(1/4).

Original entry on oeis.org

1, 1, 1, 17, 129, 2529, 42753, 1080561, 28269825, 910318785, 31733067777, 1260881785041, 54451914027393, 2588888715388065, 132887134408562433, 7371812870053439409, 437841346658159352321, 27782111830252836998529, 1873198439610729939408897
Offset: 0

Views

Author

Seiichi Manyama, Mar 05 2022

Keywords

Crossrefs

Cf. A352075, A352113.
Cf. A352073.

Programs

  • Mathematica
    m = 18; Range[0, m]! * CoefficientList[Series[(1 - Log[1 - 4*x])^(1/4), {x, 0, m}], x] (* Amiram Eldar, Mar 05 2022 *)
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace((1-log(1-4*x))^(1/4)))
  • PARI
    a(n) = sum(k=0, n, (-4)^(n-k)*prod(j=0, k-1, -4*j+1)*stirling(n, k, 1));

Formula

a(n) = Sum_{k=0..n} (-4)^(n-k) * (Product_{j=0..k-1} (-4*j+1)) * Stirling1(n,k).
a(n) ~ n! * 2^(2*n-2) / (log(n)^(3/4) * n) * (1 - 3*(gamma + 1)/(4*log(n))), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Mar 05 2022

A367429 Expansion of e.g.f. 1 / (1 - log(1 + 4*x))^(3/4).

Original entry on oeis.org

1, 3, 9, 75, 465, 7827, 54489, 1985883, 5684385, 1038408483, -8440926039, 1026884514411, -24803157926799, 1735078791616947, -69866656826056839, 4467425545047012219, -239734355869361550015, 15985164846462976491075, -1031464442408734822175415
Offset: 0

Views

Author

Seiichi Manyama, Nov 18 2023

Keywords

Crossrefs

Cf. A352073, A365600.

Programs

  • PARI
    a(n) = sum(k=0, n, 4^(n-k)*prod(j=0, k-1, 4*j+3)*stirling(n, k, 1));

Formula

a(n) = Sum_{k=0..n} 4^(n-k) * (Product_{j=0..k-1} (4*j+3)) * Stirling1(n,k).
a(0) = 1; a(n) = Sum_{k=1..n} (-4)^k * (1/4 * k/n - 1) * (k-1)! * binomial(n,k) * a(n-k).

A367426 Expansion of e.g.f. 1 / (1 + log(1 - 4*x))^(1/4).

Original entry on oeis.org

1, 1, 9, 137, 2929, 80689, 2722745, 108817785, 5028704865, 263891635425, 15505410046185, 1008591244314345, 71960155841683665, 5587928499550175505, 469183592107676627865, 42356983967876631615705, 4091474631070907136246465, 421070307443746576367920065
Offset: 0

Views

Author

Seiichi Manyama, Nov 18 2023

Keywords

Crossrefs

Cf. A352073.

Programs

  • PARI
    a(n) = sum(k=0, n, 4^(n-k)*prod(j=0, k-1, 4*j+1)*abs(stirling(n, k, 1)));

Formula

a(n) = Sum_{k=0..n} 4^(n-k) * (Product_{j=0..k-1} (4*j+1)) * |Stirling1(n,k)|.
a(0) = 1; a(n) = Sum_{k=1..n} 4^k * (1 - 3/4 * k/n) * (k-1)! * binomial(n,k) * a(n-k).
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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