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.

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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

A384332 Expansion of Product_{k>=1} (1 + k*x)^((2/3)^k).

Original entry on oeis.org

1, 6, 3, 20, -207, 2538, -36381, 599760, -11210229, 234779146, -5455240455, 139445920452, -3892724842549, 117916363928070, -3854035833235839, 135241405277665656, -5072575747811807052, 202559732310632082120, -8581116791103001216108
Offset: 0

Views

Author

Seiichi Manyama, May 26 2025

Keywords

Crossrefs

Cf. A116603, A384333, A384334.
Cf. A004123, A384324, A384344.

Programs

  • Mathematica
    terms = 20; A[] = 1; Do[A[x] = -2*A[x] + 3*A[x/(1+x)]^(2/3) * (1+x)^2 + O[x]^j // Normal, {j, 1, terms}]; CoefficientList[A[x], x] (* Vaclav Kotesovec, May 27 2025 *)
  • PARI
    my(N=20, x='x+O('x^N)); Vec(exp(3*sum(k=1, N, (-1)^(k-1)*sum(j=0, k, 2^j*j!*stirling(k, j, 2))*x^k/k)))

Formula

G.f. A(x) satisfies A(x) = (1+x)^2 * A(x/(1+x))^(2/3).
G.f.: exp(3 * Sum_{k>=1} (-1)^(k-1) * A004123(k+1) * x^k/k).
G.f.: 1/B(-x), where B(x) is the g.f. of A384324.
G.f.: B(x)^6, where B(x) is the g.f. of A384344.
a(n) ~ (-1)^(n+1) * (n-1)! / log(3/2)^(n+1). - Vaclav Kotesovec, May 27 2025

A185285 Triangle T(n,k), read by rows, given by (0, 2, 3, 4, 6, 6, 9, 8, 12, 10, 15, ...) DELTA (1, 0, 1, 0, 1, 0, 1, 0, 1, 0, ...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 10, 6, 1, 0, 74, 52, 12, 1, 0, 730, 570, 160, 20, 1, 0, 9002, 7600, 2430, 380, 30, 1, 0, 133210, 119574, 42070, 7630, 770, 42, 1, 0, 2299754, 2170252, 822696, 166320, 19740, 1400, 56, 1, 0, 45375130, 44657106, 17985268, 3956568, 528780, 44604, 2352, 72, 1
Offset: 0

Views

Author

Philippe Deléham, Dec 22 2011

Keywords

Comments

The Bell transform of A004123(n+1). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 18 2016

Examples

			Triangle begins :
1
0, 1
0, 2, 1
0, 10, 6, 1
0, 74, 52, 12, 1
0, 730, 570, 160, 20, 1
0, 9002, 7600, 2430, 380, 30, 1
0, 133210, 119574, 42070, 7630, 770, 42, 1

Crossrefs

Row sums are A136727.
Cf. A000007, A004123, A000012, A002378, A195205.

Programs

  • Mathematica
    (* The function BellMatrix is defined in A264428. *)
a4123[n_] := If[n == 1, 1, PolyLog[-n+1, 2/3]/3];
rows = 10;
M = BellMatrix[a4123[#+1]&, rows];
Table[M[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 25 2019 *)
  • Sage
    # uses[bell_matrix from A264428]
bell_matrix(lambda n: A004123(n+1), 10) # Peter Luschny, Jan 18 2016

Extensions

More terms from Jean-François Alcover, Jun 25 2019

A355233 E.g.f. A(x) satisfies A'(x) = 1 + 2 * (exp(x) - 1) * A(x).

Original entry on oeis.org

0, 1, 0, 4, 6, 40, 150, 832, 4494, 27496, 178278, 1240720, 9159678, 71523448, 588049878, 5073746464, 45800173038, 431400176008, 4230061102662, 43087882883248, 455079854567646, 4975136823055768, 56212975652894646, 655496634896272960, 7878552380411524302
Offset: 0

Views

Author

Seiichi Manyama, Jun 25 2022

Keywords

Crossrefs

Cf. A004123, A087650, A194689, A355206, A355232.

Programs

  • Mathematica
    nmax = 25; CoefficientList[Series[3*E^(-2 + 2*E^x - 2*x)/4 - 1/(E^(2*x)*4) - 1/(2*E^x), {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Jun 25 2022 *)
  • PARI
    a_vector(n) = my(v=vector(n)); v[1]=1; for(i=1, n-1, v[i+1]=2*sum(j=1, i-1, binomial(i, j)*v[j])); concat(0, v);

Formula

a(0) = 0, a(1) = 1; a(n+1) = 2 * Sum_{k=1..n-1} binomial(n,k) * a(k).
From Vaclav Kotesovec, Jun 26 2022: (Start)
E.g.f.: 3*exp(2*exp(x) - 2*x - 2)/4 - 1/(exp(2*x)*4) - 1/(2*exp(x)).
a(n) = 3*A194689(n)/4 - (-1)^n * (2^(n-2) + 1/2).
a(n) ~ 3 * n^(n-2) * exp(n/LambertW(n/2) - n - 2) / (sqrt(1 + LambertW(n/2)) * LambertW(n/2)^(n-2)). (End)

Extensions

Prepended a(0)=0 from Vaclav Kotesovec, Jun 25 2022

A365863 a(0) = 1; thereafter a(n) = n*Sum_{k = 0..n-1} binomial(n, k)*(-1)^(1+n+k)*a(k).

Original entry on oeis.org

1, 1, 2, 12, 156, 3380, 108930, 4876242, 289111032, 21916777752, 2067208751790, 237380181141950, 32601704893973556, 5276471519805880836, 993835167745129599162, 215520207875112312124890, 53311353846240820033325040, 14919977169758349265112350256, 4690364757880376663319746737926
Offset: 0

Views

Author

Thomas Scheuerle, Nov 09 2023

Keywords

Comments

Let P_k(x) be the polynomial of order k which satisfies a(m) = P_k(m) for m = 0..k, then a(k+1) = k * P_k(k+1).
This sequence is a member of a family of sequences with related properties. Here are some examples:
With b(k+1) = 1 + P_k(k+1) we get b(k) = A000079(k).
With b(k+1) = 2 + P_k(k+1) we get b(k) = A000225(k).
With b(k+1) = 3 + P_k(k+1) we get b(k) = A033484(k).
With b(k+1) = 2 * P_k(k+1) we get b(k) = A000629(k).
With b(k+1) = 1 + 2 * P_k(k+1) we get b(k) = A007047(k).
With b(k+1) = 3 * P_k(k+1) we get b(k) = A201339(k).
With b(k+1) = 5 * P_k(k+1) we get b(k) = A201365(k).
With b(k+1) = -1 * P_k(k+1) we get b(k) = A000670(k)*(-1)^k.
With b(k+1) = -2 * P_k(k+1) we get b(k) = A004123(k+1)*(-1)^k.
With b(k+1) = -3 * P_k(k+1) we get b(k) = A032033(k)*(-1)^k.
With b(k+1) = -4 * P_k(k+1) we get b(k) = A094417(k)*(-1)^k.
With b(k+1) = -m * P_k(k+1) we get b(k) = Bo(m, k)*(-1)^k, Bo(m, k) are Generalized ordered Bell numbers.

Crossrefs

Cf. A000079, A000225, A000629, A000670.
Cf. A004123, A007047, A032033, A033484.
Cf. A094417, A201339, A201365, A367308.

Programs

  • Mathematica
    a[n_] := a[n] = If[n == 0, 1, n*Sum[Binomial[n, k]*(-1)^(1 + n + k)*a[k], {k, 0, n - 1}]]; Table[a[n], {n, 0, 20}] (* Vaclav Kotesovec, Nov 12 2023 *)
  • PARI
    a(n) = if(n == 0, 1,sum(k = 0,n-1, n*binomial(n, k)*(-1)^(1+n+k)*a(k)))

Formula

a(n) ~ c * n^(2*n + 1/2) / exp(2*n), where c = 2.9711739498821842863440481701659942323709511474486414... - Vaclav Kotesovec, Nov 12 2023

A367924 Expansion of e.g.f. 1/(3 - x - 2*exp(x)).

Original entry on oeis.org

1, 3, 20, 200, 2666, 44422, 888214, 20719722, 552385386, 16567346630, 552104425070, 20238679934002, 809341290336274, 35062535546332062, 1635835480858764342, 81770970437144725034, 4360009179878123161658, 247004345719314584973430
Offset: 0

Views

Author

Seiichi Manyama, Dec 05 2023

Keywords

Crossrefs

Cf. A006155, A367925.
Cf. A004123, A367830, A367835, A367922.

Programs

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

Formula

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

A371296 E.g.f. satisfies A(x) = 1/(3 - 2*exp(x*A(x)^2)).

Original entry on oeis.org

1, 2, 26, 674, 26682, 1429682, 96867178, 7946279490, 765861255002, 84837503946962, 10621798904563530, 1483378875680954210, 228626616449674796602, 38549099486166110798322, 7058696888173770772536362, 1394913467379909728350803074, 295904373562519633314958421274
Offset: 0

Views

Author

Seiichi Manyama, Mar 18 2024

Keywords

Crossrefs

Cf. A004123, A258922.
Cf. A367134.

Programs

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

Formula

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

A382753 Expansion of e.g.f. 3/(5 - 2*exp(3*x)).

Original entry on oeis.org

1, 2, 14, 138, 1806, 29562, 580734, 13309578, 348611886, 10272416922, 336326121054, 12112707922218, 475894244100366, 20255443904321082, 928448378212678974, 45597074777924954058, 2388608236671667179246, 132947999835258872046042, 7835059049893316949502494
Offset: 0

Views

Author

Seiichi Manyama, Jun 03 2025

Keywords

Links

Crossrefs

Cf. A094417, A326324, A384435.
Cf. A004123, A216794, A384521.
Cf. A201367.

Programs

  • PARI
    a(n) = (-3)^(n+1)*polylog(-n, 5/2)/5;

Formula

a(n) = (-3)^(n+1)/5 * Li_{-n}(5/2), where Li_{n}(x) is the polylogarithm function.
a(n) = 3^(n+1)/5 * Sum_{k>=0} k^n * (2/5)^k.
a(n) = Sum_{k=0..n} 2^k * 3^(n-k) * k! * Stirling2(n,k).
a(n) = (2/5) * A201367(n) = (2/5) * Sum_{k=0..n} 5^k * (-3)^(n-k) * k! * Stirling2(n,k) for n > 0.
a(0) = 1; a(n) = 2 * Sum_{k=1..n} 3^(k-1) * binomial(n,k) * a(n-k).
a(0) = 1; a(n) = 2 * a(n-1) + 5 * Sum_{k=1..n-1} (-3)^(k-1) * binomial(n-1,k) * a(n-k).

A384412 Expansion of Product_{k>=1} 1/(1 - k^2 * x)^((1/30) * (2/3)^k).

Original entry on oeis.org

1, 1, 37, 4477, 1139503, 498101431, 332955009307, 315774077663395, 403232260150593946, 667010006578379121074, 1387375789650073950228650, 3544016332332206162590402778, 10907098996548018595779254922854, 39804369748279182675138824291484662, 169958609977149735126105997027662792638
Offset: 0

Views

Author

Seiichi Manyama, May 28 2025

Keywords

Crossrefs

Cf. A004123, A050351, A247082, A384414.

Programs

  • PARI
    b(n) = sum(k=0, n, 2^k*k!*stirling(n, k, 2));
my(N=20, x='x+O('x^N)); Vec(exp(sum(k=1, N, b(2*k)*x^k/k)/10))

Formula

G.f.: exp((1/10) * Sum_{k>=1} b(2*k) * x^k/k), where b(n) = Sum_{k=0..n} 2^k * k! * Stirling2(n,k).

A384414 Expansion of Product_{k>=1} 1/(1 - k^4 * x)^((1/30) * (2/3)^k).

Original entry on oeis.org

1, 73, 2271421, 664978095445, 805854449283423655, 2773445081734579264589407, 21807207369084946567603587345091, 339838389273170021807379637478064625867, 9495034758014772381226851471008240873743234210, 441461703234194795490537796224906335240071042475017490
Offset: 0

Views

Author

Seiichi Manyama, May 28 2025

Keywords

Crossrefs

Cf. A004123, A384412.

Programs

  • PARI
    b(n) = sum(k=0, n, 2^k*k!*stirling(n, k, 2));
my(N=10, x='x+O('x^N)); Vec(exp(sum(k=1, N, b(4*k)*x^k/k)/10))

Formula

G.f.: exp((1/10) * Sum_{k>=1} b(4*k) * x^k/k), where b(n) = Sum_{k=0..n} 2^k * k! * Stirling2(n,k).
Previous Showing 41-50 of 54 results. Next

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