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 1-3 of 3 results.

A163934 Triangle related to the asymptotic expansion of E(x,m=4,n).

Original entry on oeis.org

1, 6, 4, 35, 40, 10, 225, 340, 150, 20, 1624, 2940, 1750, 420, 35, 13132, 27076, 19600, 6440, 980, 56, 118124, 269136, 224490, 90720, 19110, 2016, 84, 1172700, 2894720, 2693250, 1265460, 330750, 48720, 3780, 120
Offset: 1

Views

Author

Johannes W. Meijer, Aug 13 2009

Keywords

Comments

The higher order exponential integrals E(x,m,n) are defined in A163931 while the general formula for their asymptotic expansion can be found in A163932.
We used the latter formula and the asymptotic expansion of E(x,m=3,n), see A163932, to determine that E(x,m=4,n) ~ (exp(-x)/x^4)*(1 - (6+4*n)/x + (35+40*n+ 10*n^2)/x^2 - (225+340*n+ 150*n^2+20*n^3)/x^3 + ... ). This formula leads to the triangle coefficients given above.
The asymptotic expansion leads for the values of n from one to five to known sequences, see the cross-references.
The numerators of the o.g.f.s. of the right hand columns of this triangle lead for z=1 to A000457, see A163939 for more information.
The first Maple program generates the sequence given above and the second program generates the asymptotic expansion of E(x,m=4,n).

Examples

			The first few rows of the triangle are:
1;
6, 4;
35, 40, 10;
225, 340, 150, 20;

Links

Crossrefs

Cf. A163931 (E(x,m,n)), A163932 and A163939.
Cf. A048994 (Stirling1), A000454 (row sums).
A000399, 4*A000454, 10*A000482, 20*A001233, 35*A001234 equal the first five left hand columns.
A000292, A027777 and A163935 equal the first three right hand columns.
The asymptotic expansion leads to A000454 (n=1), A001707 (n=2), A001713 (n=3), A001718 (n=4) and A001723 (n=5).
Cf. A130534 (m=1), A028421 (m=2), A163932 (m=3).

Programs

  • Maple
    with(combinat): A163934 := proc(n,m): (-1)^(n+m)* binomial(m+2, 3) *stirling1(n+2, m+2) end: seq(seq(A163934(n,m), m=1..n), n=1..8);
with(combinat): imax:=6; EA:=proc(x,m,n) local E, i; E:=0: for i from m-1 to imax+2 do E:=E + sum((-1)^(m+k+1)*binomial(k,m-1)*n^(k-m+1)* stirling1(i, k), k=m-1..i)/x^(i-m+1) od: E:= exp(-x)/x^(m)*E: return(E); end: EA(x,4,n);
# Maple programs revised by Johannes W. Meijer, Sep 11 2012
  • Mathematica
    a[n_, m_] /; n >= 1 && 1 <= m <= n = (-1)^(n+m)*Binomial[m+2, 3] * StirlingS1[n+2, m+2]; Flatten[Table[a[n, m], {n, 1, 8}, {m, 1, n}]][[1 ;; 36]] (* Jean-François Alcover, Jun 01 2011, after formula *)

Formula

a(n,m) = (-1)^(n+m)*C(m+2,3)*stirling1(n+2,m+2) for n >= 1 and 1<= m <= n.

A006411 Number of nonseparable tree-rooted planar maps with n + 2 edges and 3 vertices.

Original entry on oeis.org

3, 20, 75, 210, 490, 1008, 1890, 3300, 5445, 8580, 13013, 19110, 27300, 38080, 52020, 69768, 92055, 119700, 153615, 194810, 244398, 303600, 373750, 456300, 552825, 665028, 794745, 943950, 1114760, 1309440, 1530408, 1780240, 2061675, 2377620, 2731155, 3125538
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

References

  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Column 3 of A342984.
Cf. A006412, A006413, A027777.

Programs

  • Magma
    [n*(n+1)*(n+2)^2*(n+3)/24: n in [1..50]]; // Vincenzo Librandi, May 19 2011
  • Maple
    A006411:=n->n*(n+1)*(n+2)^2*(n+3)/24: seq(A006411(n), n=1..50); # Wesley Ivan Hurt, Jul 15 2017
  • Mathematica
    CoefficientList[Series[x (3+2x)/(1-x)^6,{x,0,40}],x] (* or *) LinearRecurrence[ {6,-15,20,-15,6,-1},{0,3,20,75,210,490},40] (* Harvey P. Dale, Dec 24 2013 *)

Formula

G.f.: x*(3+2*x)/(1-x)^6.
a(n) = n*(n+1)*(n+2)^2*(n+3)/24. - Bruno Berselli, May 17 2011
a(n) = A027777(n)/2. - Zerinvary Lajos, Mar 23 2007
a(n) = binomial(n+2,n)*binomial(n+2,n-1) - binomial(n+2,n+1)*binomial(n+2,n-2). - J. M. Bergot, Apr 07 2013
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6). - Harvey P. Dale, Dec 24 2013
Sum_{n>=1} 1/a(n) = 2*Pi^2 - 58/3. - Jaume Oliver Lafont, Jul 15 2017
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2 + 16*log(2) - 62/3. - Amiram Eldar, Jan 28 2022

Extensions

G.f. adapted to the offset by Bruno Berselli, May 17 2011

A163935 Third right hand column of triangle A163934.

Original entry on oeis.org

35, 340, 1750, 6440, 19110, 48720, 110880, 231000, 448305, 820820, 1431430, 2395120, 3867500, 6054720, 9224880, 13721040, 19975935, 28528500, 40042310, 55326040, 75356050, 101301200, 134550000, 176740200, 229790925, 295937460
Offset: 3

Views

Author

Johannes W. Meijer, Aug 13 2009

Keywords

Links

Crossrefs

Cf. A048994 (Stirling1).
Equals the third right hand column of triangle A163934.
A000292 and A027777 are the first and second right hand columns.

Programs

  • Maple
    nmax:=28; mmax:=nmax: with(combinat, stirling1): for n from 1 to nmax do for m from 1 to n do a(n,m):=(-1)^(n+m)*m*(m+1)*(m+2)*stirling1(n+2,m+2)/3! od; od: seq(a(n,n-2),n=3..nmax);
  • Mathematica
    CoefficientList[Series[x^3 (35 + 60 x + 10 x^2)/(1 - x)^8, {x, 0, 50}], x] (* G. C. Greubel, Aug 08 2017 *)
  • PARI
    x='x+O('x^50); Vec(x^3*(35 + 60*x + 10*x^2)/(1-x)^8) \\ G. C. Greubel, Aug 08 2017

Formula

a(n) = (n-2)*(n-1)*(n)*stirling1(n+2,n)/3!.
G.f.: x^3*(35 + 10*x^2 + 60*x)/(1 - x)^8.
Showing 1-3 of 3 results.

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

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