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 1-4 of 4 results.

A381450 Expansion of e.g.f. (1/x) * Series_Reversion( x/(1 + x * cosh(x))^3 ).

Original entry on oeis.org

1, 3, 24, 339, 7056, 195855, 6819840, 286105071, 14055420288, 791783681499, 50327779368960, 3563709848656683, 278223968271034368, 23744747385054558759, 2199369837961901789184, 219748696455778150645575, 23559108001707680103628800, 2697737574531326391439989171
Offset: 0

Views

Author

Seiichi Manyama, Feb 23 2025

Keywords

Comments

As stated in the comment of A185951, A185951(n,0) = 0^n.

Links

Crossrefs

Cf. A381171, A381449.
Cf. A377554, A381443.
Cf. A185951, A381448.

Programs

  • PARI
    a185951(n, k) = binomial(n, k)/2^k*sum(j=0, k, (2*j-k)^(n-k)*binomial(k, j));
a(n) = sum(k=0, n, k!*binomial(3*n+3, k)*a185951(n, k))/(n+1);

Formula

E.g.f. A(x) satisfies A(x) = (1 + x*A(x) * cosh(x*A(x)))^3.
E.g.f.: B(x)^3, where B(x) is the e.g.f. of A381448.
a(n) = (1/(n+1)) * Sum_{k=0..n} k! * binomial(3*n+3,k) * A185951(n,k).

A381519 Expansion of e.g.f. (1/x) * Series_Reversion( x/(1 + sin(x))^2 ).

Original entry on oeis.org

1, 2, 10, 82, 936, 13642, 240656, 4952218, 115608704, 2992207250, 84070140672, 2507383885730, 77117178496000, 2329071118971482, 61202811821836288, 690380688651775978, -88097620429234470912, -11900508444760552311518, -1112180862634722333884416
Offset: 0

Views

Author

Seiichi Manyama, Feb 26 2025

Keywords

Links

Crossrefs

Cf. A136630, A381389, A381518.
Cf. A377553, A381442, A381449, A381521.

Programs

  • PARI
    a136630(n, k) = 1/(2^k*k!)*sum(j=0, k, (-1)^(k-j)*(2*j-k)^n*binomial(k, j));
a(n) = sum(k=0, n, k!*binomial(2*n+2, k)*I^(n-k)*a136630(n, k))/(n+1);

Formula

E.g.f. A(x) satisfies A(x) = (1 + sin(x*A(x)))^2.
E.g.f.: B(x)^2, where B(x) is the e.g.f. of A381518.
a(n) = (1/(n+1)) * Sum_{k=0..n} k! * binomial(2*n+2,k) * i^(n-k) * A136630(n,k), where i is the imaginary unit.

A381478 E.g.f. A(x) satisfies A(x) = 1/( 1 - x * A(x) * cosh(x * A(x)) )^2.

Original entry on oeis.org

1, 2, 14, 186, 3696, 98290, 3283920, 132311354, 6246905728, 338374946466, 20688891816960, 1409607482926522, 105914955915952128, 8701156803022552466, 775923181679913938944, 74646655589398509637050, 7706371729268071660093440, 849834260414107910987980354
Offset: 0

Views

Author

Seiichi Manyama, Feb 24 2025

Keywords

Comments

As stated in the comment of A185951, A185951(n,0) = 0^n.

Crossrefs

Cf. A185951, A377546, A381387, A381449, A381477.

Programs

  • PARI
    a185951(n, k) = binomial(n, k)/2^k*sum(j=0, k, (2*j-k)^(n-k)*binomial(k, j));
a(n) = 2*sum(k=0, n, k!*binomial(2*n+k+2, k)/(2*n+k+2)*a185951(n, k));

Formula

E.g.f.: B(x)^2, where B(x) is the e.g.f. of A381477.
a(n) = 2 * Sum_{k=0..n} k! * binomial(2*n+k+2,k)/(2*n+k+2) * A185951(n,k).
E.g.f.: (1/x) * Series_Reversion( x*(1 - x*cosh(x))^2 ).

A381521 Expansion of e.g.f. (1/x) * Series_Reversion( x/(1 + x * cos(x))^2 ).

Original entry on oeis.org

1, 2, 10, 78, 792, 9250, 106080, 636286, -30646784, -2237508990, -112000654080, -5124930562642, -227068649702400, -9819508698442846, -406371251899045888, -15094508095346343330, -394372545425757634560, 7096803535075158290434, 2430273114806112504446976
Offset: 0

Views

Author

Seiichi Manyama, Feb 26 2025

Keywords

Comments

As stated in the comment of A185951, A185951(n,0) = 0^n.

Links

Crossrefs

Cf. A185951, A381480, A381520.
Cf. A377553, A381442, A381449, A381519.

Programs

  • PARI
    a185951(n, k) = binomial(n, k)/2^k*sum(j=0, k, (2*j-k)^(n-k)*binomial(k, j));
a(n) = sum(k=0, n, k!*binomial(2*n+2, k)*I^(n-k)*a185951(n, k))/(n+1);

Formula

E.g.f. A(x) satisfies A(x) = (1 + x*A(x) * cos(x*A(x)))^2.
E.g.f.: B(x)^2, where B(x) is the e.g.f. of A381520.
a(n) = (1/(n+1)) * Sum_{k=0..n} k! * binomial(2*n+2,k) * i^(n-k) * A185951(n,k), where i is the imaginary unit.
Showing 1-4 of 4 results.

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