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 1-7 of 7 results.

A367474 Expansion of e.g.f. 1 / (1 + 2 * log(1 - x))^2.

Original entry on oeis.org

1, 4, 28, 272, 3360, 50256, 881616, 17734944, 402278496, 10155145344, 282329361024, 8570500876032, 282047266728192, 10001430040080384, 380152962804068352, 15418451851593596928, 664633482628021493760, 30342827915683778027520
Offset: 0

Views

Author

Seiichi Manyama, Nov 19 2023

Keywords

Crossrefs

Cf. A088500, A367475.
Cf. A052801, A367470.

Programs

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

Formula

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

A367471 Expansion of e.g.f. 1 / (3 - 2 * exp(x))^3.

Original entry on oeis.org

1, 6, 54, 630, 8982, 150966, 2918934, 63772470, 1552910742, 41690570166, 1223096629014, 38924237638710, 1335418262833302, 49129420920630966, 1929262811804022294, 80540656071983191350, 3561781875173605408662, 166331104582900651581366
Offset: 0

Views

Author

Seiichi Manyama, Nov 19 2023

Keywords

Crossrefs

Cf. A004123, A367470.
Cf. A226515, A367473.

Programs

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

Formula

a(n) = (1/2) * Sum_{k=0..n} 2^k * (k+2)! * Stirling2(n,k).
a(0) = 1; a(n) = 2*Sum_{k=1..n} (2*k/n + 1) * binomial(n,k) * a(n-k).
a(0) = 1; a(n) = 6*a(n-1) - 3*Sum_{k=1..n-1} (-1)^k * binomial(n-1,k) * a(n-k).

A367472 Expansion of e.g.f. 1 / (4 - 3 * exp(x))^2.

Original entry on oeis.org

1, 6, 60, 816, 13992, 289176, 6990360, 193432056, 6028092312, 208891033656, 7966989308760, 331618933474296, 14958464943057432, 726825458489514936, 37846457287387667160, 2102428978611587164536, 124109778776508893651352, 7758254465575303379273016
Offset: 0

Views

Author

Seiichi Manyama, Nov 19 2023

Keywords

Crossrefs

Cf. A032033, A367473.
Cf. A005649, A367470.

Programs

  • PARI
    a(n) = sum(k=0, n, 3^k*(k+1)!*stirling(n, k, 2));

Formula

a(n) = Sum_{k=0..n} 3^k * (k+1)! * Stirling2(n,k).
a(0) = 1; a(n) = 3*Sum_{k=1..n} (k/n + 1) * binomial(n,k) * a(n-k).
a(0) = 1; a(n) = 6*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

A375991 Expansion of e.g.f. (3 - 2 * exp(x))^(3/2).

Original entry on oeis.org

1, -3, 0, 9, 45, 252, 1935, 19989, 260190, 4063887, 73823445, 1527002694, 35408499885, 909389617497, 25618701424680, 785355764569749, 26024092206299505, 926859918577582332, 35306305954587340515, 1432301360556686816529, 61649353087003554947550
Offset: 0

Views

Author

Seiichi Manyama, Sep 05 2024

Keywords

Crossrefs

Cf. A004123, A305404, A367470, A375948, A375954.
Cf. A006681.

Programs

  • Mathematica
    With[{nn=20},CoefficientList[Series[(3-2Exp[x])^(3/2),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, May 19 2025 *)
  • PARI
    a(n) = sum(k=0, n, prod(j=0, k-1, 2*j-3)*stirling(n, k, 2));

Formula

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

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