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.

A367470 Expansion of e.g.f. 1 / (3 - 2 * exp(x))^2.

Original entry on oeis.org

1, 4, 28, 268, 3244, 47404, 810988, 15891628, 350851564, 8615761324, 232911898348, 6872755977388, 219799913877484, 7572909749244844, 279630706025296108, 11016315458773541548, 461211305514352065004, 20448268640012928321964
Offset: 0

Views

Author

Seiichi Manyama, Nov 19 2023

Keywords

Crossrefs

Cf. A004123, A367471.
Cf. A005649, A367472.
Cf. A367474.

Programs

  • PARI
    a(n) = sum(k=0, n, 2^k*(k+1)!*stirling(n, k, 2));

Formula

a(n) = Sum_{k=0..n} 2^k * (k+1)! * Stirling2(n,k).
a(0) = 1; a(n) = 2*Sum_{k=1..n} (k/n + 1) * binomial(n,k) * a(n-k).
a(0) = 1; a(n) = 4*a(n-1) - 3*Sum_{k=1..n-1} (-1)^k * binomial(n-1,k) * a(n-k).
a(n) ~ sqrt(2*Pi) * n^(n + 3/2) / (9 * log(3/2)^(n+2) * exp(n)). - Vaclav Kotesovec, May 20 2025

A367475 Expansion of e.g.f. 1 / (1 + 2 * log(1 - x))^3.

Original entry on oeis.org

1, 6, 54, 636, 9204, 157584, 3111312, 69533472, 1734229344, 47733263232, 1436801816448, 46942939272960, 1654215709835520, 62533593070755840, 2524077593084160000, 108339176213529384960, 4927173048408858531840, 236673892535088351744000
Offset: 0

Views

Author

Seiichi Manyama, Nov 19 2023

Keywords

Crossrefs

Cf. A088500, A367474.
Cf. A354122, A367471.

Programs

  • Maple
    A367475 := proc(n)
    option remember ;
    if n =0 then
        1;
    else
        2*add((2*k/n + 1) * (k-1)! * binomial(n,k) * procname(n-k),k=1..n) ;
    end if;
end proc:
seq(A367475(n),n=0..70) ; # R. J. Mathar, Dec 04 2023
  • PARI
    a(n) = sum(k=0, n, 2^k*(k+2)!*abs(stirling(n, k, 1)))/2;

Formula

a(n) = (1/2) * Sum_{k=0..n} 2^k * (k+2)! * |Stirling1(n,k)|.
a(0) = 1; a(n) = 2*Sum_{k=1..n} (2*k/n + 1) * (k-1)! * binomial(n,k) * a(n-k).

A367473 Expansion of e.g.f. 1 / (4 - 3 * exp(x))^3.

Original entry on oeis.org

1, 9, 117, 1953, 39645, 946089, 25926597, 801869553, 27618402285, 1048096422009, 43444114011477, 1952712851250753, 94592798546953725, 4912513525545837129, 272265236648295312357, 16039329591716508497553, 1000809252891040145821965
Offset: 0

Views

Author

Seiichi Manyama, Nov 19 2023

Keywords

Crossrefs

Cf. A032033, A367472.
Cf. A226515, A367471.

Programs

  • PARI
    a(n) = sum(k=0, n, 3^k*(k+2)!*stirling(n, k, 2))/2;

Formula

a(n) = (1/2) * Sum_{k=0..n} 3^k * (k+2)! * Stirling2(n,k).
a(0) = 1; a(n) = 3*Sum_{k=1..n} (2*k/n + 1) * binomial(n,k) * a(n-k).
a(0) = 1; a(n) = 9*a(n-1) - 4*Sum_{k=1..n-1} (-1)^k * binomial(n-1,k) * a(n-k).

A375948 Expansion of e.g.f. 1 / (3 - 2 * exp(x))^(3/2).

Original entry on oeis.org

1, 3, 18, 153, 1683, 22698, 362403, 6683463, 139787568, 3269240883, 84535585263, 2394699999948, 73749495626253, 2453332830142743, 87667856626175298, 3349116499958627733, 136209377351085310863, 5875794769594996985778, 267968680043585007829383
Offset: 0

Views

Author

Seiichi Manyama, Sep 03 2024

Keywords

Crossrefs

Cf. A005649, A375949.
Cf. A004123, A367470, A367471, A375954.
Cf. A001147.

Programs

  • Mathematica
    nmax=18; CoefficientList[Series[1 / (3 - 2 * Exp[x])^(3/2),{x,0,nmax}],x]*Range[0,nmax]! (* Stefano Spezia, Sep 03 2024 *)
  • PARI
    a001147(n) = prod(k=0, n-1, 2*k+1);
a(n) = sum(k=0, n, a001147(k+1)*stirling(n, k, 2));

Formula

a(n) = Sum_{k=0..n} A001147(k+1) * Stirling2(n,k).
a(n) ~ 2^(3/2) * n^(n+1) / (3^(3/2) * log(3/2)^(n + 3/2) * exp(n)). - Vaclav Kotesovec, May 20 2025

A367486 Expansion of e.g.f. 1/(3 - 2*exp(x))^x.

Original entry on oeis.org

1, 0, 4, 18, 168, 1830, 24540, 388122, 7084560, 146650446, 3395460900, 86962122786, 2441210321880, 74542218945558, 2459830123779756, 87236196407090730, 3308881779086345760, 133667058288336876894, 5729380391745420070068
Offset: 0

Views

Author

Seiichi Manyama, Nov 19 2023

Keywords

Crossrefs

Cf. A004123, A367470, A367471.
Cf. A354413, A367488.
Cf. A367489.

Programs

  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, j*sum(k=1, j-1, 2^k*(k-1)!*stirling(j-1, k, 2))*binomial(i-1, j-1)*v[i-j+1])); v;

Formula

a(0) = 1; a(n) = Sum_{k=1..n} A367489(k) * binomial(n-1,k-1) * a(n-k).

A375954 Expansion of e.g.f. 1 / (3 - 2 * exp(x))^(5/2).

Original entry on oeis.org

1, 5, 40, 425, 5605, 88100, 1606015, 33291725, 773093830, 19875432575, 560334083965, 17187010139150, 569768238573805, 20299523526975425, 773470729977309040, 31385122689116278325, 1351135296804805544905, 61507193821772778512900
Offset: 0

Views

Author

Seiichi Manyama, Sep 03 2024

Keywords

Crossrefs

Cf. A004123, A367470, A367471, A375948.
Cf. A001147.

Programs

  • Mathematica
    nmax=17; CoefficientList[Series[1 / (3 - 2 * Exp[x])^(5/2),{x,0,nmax}],x]*Range[0,nmax]! (* Stefano Spezia, Sep 03 2024 *)
  • PARI
    a001147(n) = prod(k=0, n-1, 2*k+1);
a(n) = sum(k=0, n, a001147(k+2)*stirling(n, k, 2))/3;

Formula

a(n) = (1/3) * Sum_{k=0..n} A001147(k+2) * Stirling2(n,k).
a(n) ~ 2^(5/2) * n^(n+2) / (3^(7/2) * log(3/2)^(n + 5/2) * exp(n)). - Vaclav Kotesovec, May 20 2025
Showing 1-6 of 6 results.

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