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 1-2 of 2 results.

A352070 Expansion of e.g.f. 1/(1 - log(1 + 3*x))^(1/3).

Original entry on oeis.org

1, 1, 1, 10, 10, 604, -1844, 107344, -1201400, 42193576, -875584376, 29853569008, -880141783184, 32865860907424, -1216481572723616, 51296026356128512, -2244334822166729600, 106984479644794783360, -5358207684820194270080, 286466413246622566048000
Offset: 0

Views

Author

Seiichi Manyama, Mar 05 2022

Keywords

Crossrefs

Cf. A006252, A097397, A352073.
Cf. A007559, A133480, A352113, A352118.

Programs

  • Mathematica
    m = 19; Range[0, m]! * CoefficientList[Series[(1 - Log[1 + 3*x])^(-1/3), {x, 0, m}], x] (* Amiram Eldar, Mar 05 2022 *)
Table[Sum[3^(n-k) * Product[3*j+1, {j,0,k-1}] * StirlingS1[n,k], {k,0,n}], {n,0,20}] (* Vaclav Kotesovec, Sep 07 2023 *)
  • Maxima
    a[n]:=if n=0 then 1 else n!*sum(a[n-k]*(2/n/3-1/k)*(-3)^k/(n-k)!,k,1,n);
makelist(a[n],n,0,50); /* Tani Akinari, Sep 07 2023 */
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1-log(1+3*x))^(1/3)))
  • PARI
    a(n) = sum(k=0, n, 3^(n-k)*prod(j=0, k-1, 3*j+1)*stirling(n, k, 1));

Formula

a(n) = Sum_{k=0..n} 3^(n-k) * (Product_{j=0..k-1} (3*j+1)) * Stirling1(n,k).
For n > 0, a(n) = n!*Sum_{k=1..n} a(n-k)*(2/n/3-1/k)*(-3)^k/(n-k)!. - Tani Akinari, Sep 07 2023
a(n) ~ -(-1)^n * 3^(n-1) * n! / (n * log(n)^(4/3)) * (1 - 4*(1+gamma)/(3*log(n))), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Sep 07 2023

A136215 Triangle T, read by rows, where T(n,k) = A007559(n-k)*C(n,k) where A007559 equals the triple factorials in column 0.

Original entry on oeis.org

1, 1, 1, 4, 2, 1, 28, 12, 3, 1, 280, 112, 24, 4, 1, 3640, 1400, 280, 40, 5, 1, 58240, 21840, 4200, 560, 60, 6, 1, 1106560, 407680, 76440, 9800, 980, 84, 7, 1, 24344320, 8852480, 1630720, 203840, 19600, 1568, 112, 8, 1, 608608000, 219098880, 39836160
Offset: 0

Views

Author

Paul D. Hanna, Feb 07 2008

Keywords

Comments

Comments from Peter Bala, Jul 10 2008: (Start) This array is the particular case P(1,3) of the generalized Pascal triangle P(a,b), a lower unit triangular matrix, shown below
n\k|0....................1...............2.........3.....4
----------------------------------------------------------
0..|1.....................................................
1..|a....................1................................
2..|a(a+b)...............2a..............1................
3..|a(a+b)(a+2b).........3a(a+b).........3a........1......
4..|a(a+b)(a+2b)(a+3b)...4a(a+b)(a+2b)...6a(a+b)...4a....1
...
See A094587 for some general properties of these arrays.
Other cases recorded in the database include: P(1,0) = Pascal's triangle A007318, P(1,1) = A094587, P(2,0) = A038207, P(3,0) = A027465, P(2,1) = A132159 and P(2,3) = A136216. (End)
The generalized Pascal matrix that Bala refers to is itself a special case of application of the formalism of A133314 to fundamental matrices derived from infinitesimal generators described in A133314, of which the fundamental Pascal (A007318), unsigned Lah (A105278) and associated Laguerre (A135278) matrices are special examples. The formalism gives, among other relations, the inverse of T as TI(n,k) = b(n-k)*C(n,k) where the sequence b is given by the list partition transform (A133314) of A007559; i.e., b = LPT(A007559) = (1,-A008544)= (1,-1,-2,-10,-80,...). The formalism of A132382 may also be applied with the double factorial A001147 replaced by the triple factorial A007559 (see also A133480). - Tom Copeland, Aug 18 2008
From Peter Bala, Aug 29 2013: (Start)
Exponential Riordan array [1/(1 - 3*y)^(1/3), y]. The row polynomials R(n,x) thus form a Sheffer sequence of polynomials with associated delta operator equal to d/dx. Thus d/dx(R(n,x)) = n*R(n-1,x). The Sheffer identity is R(n,x + y) = sum {k = 0..n} binomial(n,k)*y^(n-k)*R(k,x).
Define a polynomial sequence P(n,x) of binomial type by setting P(n,x) = product {k = 0..n-1} (x + 3*k) with the convention that P(0,x) = 1. Then this is triangle of connection constants when expressing the basis polynomials P(n,x + 1) in terms of the basis P(n,x).
For example, row 3 is (28, 12, 3, 1) so P(3,x + 1) = (x + 1)*(x + 4)*(x + 7) = 28 + 12*x + 3*x*(x + 3) + x*(x + 3)*(x + 6). (End)

Examples

			Column k of T = column 0 of U^(k+1), while
column k of U = column 0 of T^(3k+1) where U = A136214 and
column k of V = column 0 of T^(3k+2) where V = A112333.
This triangle T begins:
        1;
        1,      1;
        4,      2,     1;
       28,     12,     3,    1;
      280,    112,    24,    4,   1;
     3640,   1400,   280,   40,   5,  1;
    58240,  21840,  4200,  560,  60,  6, 1;
  1106560, 407680, 76440, 9800, 980, 84, 7, 1; ...
Triangle U = A136214 begins:
     1;
     1,    1;
     4,    4,   1;
    28,   28,   7,   1;
   280,  280,  70,  10,  1;
  3640, 3640, 910, 130, 13, 1; ...
with triple factorials A007559 in column 0.
Triangle V = A112333 begins:
      1;
      2,    1;
     10,    5,    1;
     80,   40,    8,   1;
    880,  440,   88,  11,  1;
  12320, 6160, 1232, 154, 14, 1; ...
with triple factorials A008544 in column 0.

Links

Crossrefs

Cf. A136216 (matrix square); A007559, A008544; A136212, A136213.
Cf. A094587.

Programs

  • Mathematica
    T[n_, k_]:= Binomial[n, k]*If[n - k == 0, 1, Product[3*j + 1, {j, 0, n - k - 1}]]; Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Jun 10 2018 *)
  • PARI
    T(n,k)=binomial(n,k)*if(n-k==0,1,prod(j=0,n-k-1,3*j+1))

Formula

Column k of T = column 0 of U^(k+1) (matrix power) for k>=0 where U = A136214. Matrix square equals A136216, where A136216(n,k) = A008544(n-k)*C(n,k) where A008544 are also triple factorials.
From Peter Bala, Jul 10 2008: (Start)
T(n,k) = (3*n-3*k-2)*T(n-1,k) + T(n-1,k-1).
E.g.f. exp(x*y)/(1-3*y)^(1/3) = 1 + (1+x)*y + (4+2*x+x^2)*y^2/2! + ... . (End)
Showing 1-2 of 2 results.

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

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