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.

Previous Showing 11-20 of 24 results. Next

A351733 Expansion of e.g.f. exp( 2 * x * (exp(x) - 1) ).

Original entry on oeis.org

1, 0, 4, 6, 56, 250, 1812, 12614, 101040, 864882, 7988780, 78726142, 823897032, 9111774698, 106068603396, 1295153135670, 16538681229152, 220281968528098, 3053087839536732, 43941561067048430, 655501502129291640, 10118103843683127642
Offset: 0

Views

Author

Seiichi Manyama, May 20 2022

Keywords

Crossrefs

Cf. A052506, A351734.
Cf. A053489, A351736.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(2*x*(exp(x)-1))))
  • PARI
    a(n) = n!*sum(k=0, n\2, 2^k*stirling(n-k, k, 2)/(n-k)!);

Formula

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

A351734 Expansion of e.g.f. exp( 3 * x * (exp(x) - 1) ).

Original entry on oeis.org

1, 0, 6, 9, 120, 555, 5148, 39711, 378528, 3715011, 39838260, 452684463, 5463506304, 69553644771, 930940368036, 13054086036855, 191222363275968, 2918620069099395, 46309955947643124, 762335523354333855, 12995722456718984160, 229045407317491457763
Offset: 0

Views

Author

Seiichi Manyama, May 20 2022

Keywords

Crossrefs

Cf. A052506, A351733.
Cf. A053490, A351737.

Programs

  • Mathematica
    With[{nn=30},CoefficientList[Series[Exp[3x (Exp[x]-1)],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Apr 02 2025 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(3*x*(exp(x)-1))))
  • PARI
    a(n) = n!*sum(k=0, n\2, 3^k*stirling(n-k, k, 2)/(n-k)!);

Formula

a(n) = n! * Sum_{k=0..floor(n/2)} 3^k * Stirling2(n-k,k)/(n-k)!.

A353894 Expansion of e.g.f. exp( (x * (exp(x) - 1))^2 / 4 ).

Original entry on oeis.org

1, 0, 0, 0, 6, 30, 105, 315, 2128, 24948, 251415, 2093025, 16437036, 148728294, 1693067467, 21459867975, 270217289280, 3338860150488, 42428729660751, 581966068060485, 8654787480759700, 135253842794286930, 2163416823356628147, 35313421249845594075
Offset: 0

Views

Author

Seiichi Manyama, May 09 2022

Keywords

Crossrefs

Cf. A052506, A353895, A353896.
Cf. A060311, A353883, A353891.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp((x*(exp(x)-1))^2/4)))
  • PARI
    a(n) = n!*sum(k=0, n\4, (2*k)!*stirling(n-2*k, 2*k, 2)/(4^k*k!*(n-2*k)!));

Formula

a(n) = n! * Sum_{k=0..floor(n/4)} (2*k)! * Stirling2(n-2*k,2*k)/(4^k * k! * (n-2*k)!).

A353895 Expansion of e.g.f. exp( (x * (exp(x) - 1))^3 / 36 ).

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 20, 210, 1400, 7560, 36120, 159390, 850300, 9875580, 170133964, 2688015330, 36706233200, 444802722000, 4939264076016, 52543545234534, 583037908936500, 7645631225897700, 124931080233222340, 2327407301807577066, 44282377224446369800
Offset: 0

Views

Author

Seiichi Manyama, May 09 2022

Keywords

Crossrefs

Cf. A052506, A353894, A353896.
Cf. A327504, A353884, A353892.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp((x*(exp(x)-1))^3/36)))
  • PARI
    a(n) = n!*sum(k=0, n\6, (3*k)!*stirling(n-3*k, 3*k, 2)/(36^k*k!*(n-3*k)!));

Formula

a(n) = n! * Sum_{k=0..floor(n/6)} (3*k)! * Stirling2(n-3*k,3*k)/(36^k * k! * (n-3*k)!).

A353896 Expansion of e.g.f. exp( (x * (exp(x) - 1))^4 / 576 ).

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 70, 1260, 13650, 115500, 841995, 5555550, 34139105, 198948750, 1144463320, 8171563400, 112204064700, 2364061354200, 49912312090845, 951208121086650, 16403948060775275, 260328078068154250, 3860274855288458376, 54182965918066177500
Offset: 0

Views

Author

Seiichi Manyama, May 09 2022

Keywords

Crossrefs

Cf. A052506, A353894, A353895.
Cf. A327505, A353885, A353893.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp((x*(exp(x)-1))^4/576)))
  • PARI
    a(n) = n!*sum(k=0, n\8, (4*k)!*stirling(n-4*k, 4*k, 2)/(576^k*k!*(n-4*k)!));

Formula

a(n) = n! * Sum_{k=0..floor(n/8)} (4*k)! * Stirling2(n-4*k,4*k)/(576^k * k! * (n-4*k)!).

A354311 Expansion of e.g.f. exp( x/2 * (exp(2 * x) - 1) ).

Original entry on oeis.org

1, 0, 2, 6, 28, 160, 1056, 7784, 63568, 569664, 5542240, 58038112, 650045760, 7746901760, 97790608384, 1302349549440, 18235836899584, 267663541270528, 4107395264113152, 65739857693144576, 1095095457262013440, 18949711553467957248, 340036076121127395328
Offset: 0

Views

Author

Seiichi Manyama, May 23 2022

Keywords

Crossrefs

Cf. A052506, A354312.
Cf. A351733, A351736, A354309, A354313.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x/2*(exp(2*x)-1))))
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=2, i, j*2^(j-2)*binomial(i-1, j-1)*v[i-j+1])); v;
  • PARI
    a(n) = n!*sum(k=0, n\2, 2^(n-2*k)*stirling(n-k, k, 2)/(n-k)!);

Formula

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

A356798 E.g.f. satisfies log(A(x)) = x * (exp(x) - 1) * A(x)^3.

Original entry on oeis.org

1, 0, 2, 3, 88, 425, 13476, 130417, 4543120, 71005041, 2723297860, 60685651961, 2564091428856, 75166650583609, 3496499475113932, 127585829832674865, 6521845096842043936, 284745004488498858209, 15950013722559213419412, 809403234909367349670409
Offset: 0

Views

Author

Seiichi Manyama, Aug 28 2022

Keywords

Links

Crossrefs

Cf. A052506, A355843, A356797.
Cf. A356789.

Programs

  • Mathematica
    nmax = 19; A[_] = 1;
Do[A[x_] = Exp[x*(Exp[x] - 1)*A[x]^3] + O[x]^(nmax+1) // Normal, {nmax}];
CoefficientList[A[x], x]*Range[0, nmax]! (* Jean-François Alcover, Mar 05 2024 *)
  • PARI
    a(n) = n!*sum(k=0, n\2, (3*k+1)^(k-1)*stirling(n-k, k, 2)/(n-k)!);
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (3*k+1)^(k-1)*(x*(exp(x)-1))^k/k!)))
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(3*x*(1-exp(x)))/3)))
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace((lambertw(3*x*(1-exp(x)))/(3*x*(1-exp(x))))^(1/3)))

Formula

a(n) = n! * Sum_{k=0..floor(n/2)} (3*k+1)^(k-1) * Stirling2(n-k,k)/(n-k)!.
E.g.f.: A(x) = Sum_{k>=0} (3*k+1)^(k-1) * (x * (exp(x) - 1))^k / k!.
E.g.f.: A(x) = exp( -LambertW(3 * x * (1 - exp(x)))/3 ).
E.g.f.: A(x) = ( LambertW(3 * x * (1 - exp(x)))/(3 * x * (1 - exp(x))) )^(1/3).

A354312 Expansion of e.g.f. exp( x/3 * (exp(3 * x) - 1) ).

Original entry on oeis.org

1, 0, 2, 9, 48, 315, 2496, 22491, 223728, 2437371, 28931040, 371291283, 5111412120, 75014135235, 1168157451384, 19228202401635, 333378840718944, 6069073767712587, 115683487658404272, 2303091818149762899, 47784447190060311240, 1031179733234906055507
Offset: 0

Views

Author

Seiichi Manyama, May 23 2022

Keywords

Crossrefs

Cf. A052506, A354311.
Cf. A351734, A351737, A354310, A354314.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x/3*(exp(3*x)-1))))
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=2, i, j*3^(j-2)*binomial(i-1, j-1)*v[i-j+1])); v;
  • PARI
    a(n) = n!*sum(k=0, n\2, 3^(n-2*k)*stirling(n-k, k, 2)/(n-k)!);

Formula

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

A362369 Triangle read by rows, T(n, k) = binomial(n, k) * k! * Stirling2(n-k, k), for n >= 0 and 0 <= k <= n//2, where '//' denotes integer division.

Original entry on oeis.org

1, 0, 0, 2, 0, 3, 0, 4, 12, 0, 5, 60, 0, 6, 210, 120, 0, 7, 630, 1260, 0, 8, 1736, 8400, 1680, 0, 9, 4536, 45360, 30240, 0, 10, 11430, 216720, 327600, 30240, 0, 11, 28050, 956340, 2772000, 831600, 0, 12, 67452, 3993000, 20207880, 13305600, 665280
Offset: 0

Views

Author

Peter Luschny, May 04 2023

Keywords

Examples

			Triangle T(n, k) starts:
[0] 1;
[1] 0;
[2] 0, 2;
[3] 0, 3;
[4] 0, 4,   12;
[5] 0, 5,   60;
[6] 0, 6,  210,   120;
[7] 0, 7,  630,  1260;
[8] 0, 8, 1736,  8400,  1680;
[9] 0, 9, 4536, 45360, 30240;

Crossrefs

Cf. A000169, A052506 (row sums), A362788, A362789.

Programs

  • Maple
    T := (n, k) -> binomial(n, k) * k! * Stirling2(n-k, k):
seq(seq(T(n, k), k = 0..iquo(n, 2)), n = 0..9);
# second program:
egf := k-> (x*(exp(x)-1))^k / k!:
A362369 := (n, k)-> n! * coeff(series(egf(k), x, n+1), x, n):
seq(print(seq(A362369(n, k), k=0..iquo(n,2))), n=0..12); # Mélika Tebni, May 10 2023
  • SageMath
    def A362369(n, k):
    return binomial(n, k) * factorial(k) * stirling_number2(n - k, k)
for n in range(10):
    print([A362369(n, k) for k in range(n//2 + 1)])

Formula

From Mélika Tebni, May 10 2023: (Start)
E.g.f. of column k: (x*(exp(x)-1))^k / k!.
Sum_{k=0..n-1} (-1)^(n+k-1)*T(n+k-1, k) = A000169(n), for n > 0. (End)

A361652 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..floor(n/2)} k^j * Stirling2(n-j,j)/(n-j)!.

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 4, 3, 0, 1, 0, 6, 6, 16, 0, 1, 0, 8, 9, 56, 65, 0, 1, 0, 10, 12, 120, 250, 336, 0, 1, 0, 12, 15, 208, 555, 1812, 1897, 0, 1, 0, 14, 18, 320, 980, 5148, 12614, 11824, 0, 1, 0, 16, 21, 456, 1525, 11064, 39711, 101040, 80145, 0
Offset: 0

Views

Author

Seiichi Manyama, May 05 2023

Keywords

Examples

			Square array begins:
  1,  1,   1,   1,   1,    1, ...
  0,  0,   0,   0,   0,    0, ...
  0,  2,   4,   6,   8,   10, ...
  0,  3,   6,   9,  12,   15, ...
  0, 16,  56, 120, 208,  320, ...
  0, 65, 250, 555, 980, 1525, ...

Crossrefs

Columns k=0..3 give: A000007, A052506, A351733, A351734.
Main diagonal gives (-1)^n * A290158(n).
Cf. A362834.

Programs

  • PARI
    T(n, k) = n!*sum(j=0, n\2, k^j*stirling(n-j, j, 2)/(n-j)!);

Formula

E.g.f. of column k: exp(k * x * (exp(x) - 1)).
Previous Showing 11-20 of 24 results. Next

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

Content is available under The OEIS End-User License Agreement.