A049388 -id:A049388 - cp's OEIS frontend

A172455 The case S(6,-4,-1) of the family of self-convolutive recurrences studied by Martin and Kearney.

1, 7, 84, 1463, 33936, 990542, 34938624, 1445713003, 68639375616, 3676366634402, 219208706540544, 14397191399702118, 1032543050697424896, 80280469685284582812, 6725557192852592984064, 603931579625379293509683

Offset: 1

N. J. A. Sloane, Nov 20 2010

nonn

			G.f. = x + 7*x^2 + 84*x^3 + 1463*x^4 + 33936*x^5 + 990542*x^6 + 34938624*x^7 + ...
a(2) = 7 since (6*2 - 4) * a(2-1) - (a(1) * a(2-1)) = 7.

Vincenzo Librandi, Table of n, a(n) for n = 1..200
R. J. Martin and M. J. Kearney, An exactly solvable self-convolutive recurrence, Aequat. Math., 80 (2010), 291-318. see p. 307.
R. J. Martin and M. J. Kearney, An exactly solvable self-convolutive recurrence, arXiv:1103.4936 [math.CO], 2011.
NIST Digital Library of Mathematical Functions, Airy Functions.
A. N. Stokes, Continued fraction solutions of the Riccati equation, Bull. Austral. Math. Soc. Vol. 25 (1982), 207-214.
Eric Weisstein's World of Mathematics, Airy Functions, contains the definitions of Ai(x), Bi(x).

Cf. A000079 S(1,1,-1), A000108 S(0,0,1), A000142 S(1,-1,0), A000244 S(2,1,-2), A000351 S(4,1,-4), A000400 S(5,1,-5), A000420 S(6,1,-6), A000698 S(2,-3,1), A001710 S(1,1,0), A001715 S(1,2,0), A001720 S(1,3,0), A001725 S(1,4,0), A001730 S(1,5,0), A003319 S(1,-2,1), A005411 S(2,-4,1), A005412 S(2,-2,1), A006012 S(-1,2,2), A006318 S(0,1,1), A047891 S(0,2,1), A049388 S(1,6,0), A051604 S(3,1,0), A051605 S(3,2,0), A051606 S(3,3,0), A051607 S(3,4,0), A051608 S(3,5,0), A051609 S(3,6,0), A051617 S(4,1,0), A051618 S(4,2,0), A051619 S(4,3,0), A051620 S(4,4,0), A051621 S(4,5,0), A051622 S(4,6,0), A051687 S(5,1,0), A051688 S(5,2,0), A051689 S(5,3,0), A051690 S(5,4,0), A051691 S(5,5,0), A053100 S(6,1,0), A053101 S(6,2,0), A053102 S(6,3,0), A053103 S(6,4,0), A053104 S(7,1,0), A053105 S(7,2,0), A053106 S(7,3,0), A062980 S(6,-8,1), A082298 S(0,3,1), A082301 S(0,4,1), A082302 S(0,5,1), A082305 S(0,6,1), A082366 S(0,7,1), A082367 S(0,8,1), A105523 S(0,-2,1), A107716 S(3,-4,1), A111529 S(1,-3,2), A111530 S(1,-4,3), A111531 S(1,-5,4), A111532 S(1,-6,5), A111533 S(1,-7,6), A111546 S(1,0,1), A111556 S(1,1,1), A143749 S(0,10,1), A146559 S(1,1,-2), A167872 S(2,-3,2), A172450 S(2,0,-1), A172485 S(-1,-2,3), A177354 S(1,2,1), A292186 S(4,-6,1), A292187 S(3, -5, 1).

a[1] = 1; a[n_]:= a[n] = (6*n-4)*a[n-1] - Sum[a[k]*a[n-k], {k, 1, n-1}]; Table[a[n], {n, 1, 20}] (* Vaclav Kotesovec, Jan 19 2015 *)

{a(n) = local(A); if( n<1, 0, A = vector(n); A[1] = 1; for( k=2, n, A[k] = (6 * k - 4) * A[k-1] - sum( j=1, k-1, A[j] * A[k-j])); A[n])} /* Michael Somos, Jul 24 2011 */

S(v1, v2, v3, N=16) = {
  my(a = vector(N)); a[1] = 1;
  for (n = 2, N, a[n] = (v1*n+v2)*a[n-1] + v3*sum(j=1,n-1,a[j]*a[n-j])); a;
};
S(6,-4,-1)
\\ test: y = x*Ser(S(6,-4,-1,201)); 6*x^2*y' == y^2 - (2*x-1)*y - x
\\ Gheorghe Coserea, May 12 2017

a(n) = (6*n - 4) * a(n-1) - Sum_{k=1..n-1} a(k) * a(n-k) if n>1. - Michael Somos, Jul 24 2011
G.f.: x / (1 - 7*x / (1 - 5*x / (1 - 13*x / (1 - 11*x / (1 - 19*x / (1 - 17*x / ... )))))). - Michael Somos, Jan 03 2013
a(n) = 3/(2*Pi^2)*int((4*x)^((3*n-1)/2)/(Ai'(x)^2+Bi'(x)^2), x=0..inf), where Ai'(x), Bi'(x) are the derivatives of the Airy functions. [Vladimir Reshetnikov, Sep 24 2013]
a(n) ~ 6^n * (n-1)! / (2*Pi) [Martin + Kearney, 2011, p.16]. - Vaclav Kotesovec, Jan 19 2015
6*x^2*y' = y^2 - (2*x-1)*y - x, where y(x) = Sum_{n>=1} a(n)*x^n. - Gheorghe Coserea, May 12 2017
G.f.: x/(1 - 2*x - 5*x/(1 - 7*x/(1 - 11*x/(1 - 13*x/(1 - ... - (6*n - 1)*x/(1 - (6*n + 1)*x/(1 - .... Cf. A062980. - Peter Bala, May 21 2017

A303614 a(n) = [x^n] (1/5040 * Sum_{k=0..n} (k+7)!*x^k)^(1/4).

Original entry on oeis.org

1, 2, 12, 100, 1016, 11904, 155896, 2237264, 34713216, 576780384, 10190672448, 190437245568, 3748465227360, 77458688700864, 1675826429735808, 37874833476394368, 892480151181131520, 21889831041022132224, 557997984025699216128

Offset: 0

Seiichi Manyama, Apr 27 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..444

Cf. A049388, A303569.

N=66; x='x+O('x^N); Vec((1/7!*sum(k=0, N, (k+7)!*x^k))^(1/4))

A051561 Third unsigned column of triangle A051379.

Original entry on oeis.org

0, 0, 1, 27, 539, 9850, 176554, 3197348, 59354028, 1137868848, 22614500016, 466814750688, 10015620672672, 223359393479040, 5175622796192640, 124533006364442880, 3109120944743427840, 80473740053567016960

Offset: 0

Wolfdieter Lang

From Johannes W. Meijer, Oct 20 2009: (Start)
The asymptotic expansion of the higher order exponential integral E(x,m=3,n=8) ~ exp(-x)/x^3*(1 - 27/x + 539/x^2 - 9850/x^3 + 176554/x^4 + ...) leads to the sequence given above. See A163931 and A163932 for more information.
(End)

Mitrinovic, D. S. and Mitrinovic, R. S. see reference given for triangle A051379.

Cf. A049388 (m=0), A051560 (m=1) unsigned columns.

With[{nn=20},CoefficientList[Series[(Log[1-x])^2/(2(1-x)^8),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jul 10 2013 *)

a(n) = A051379(n, 2)*(-1)^n; e.g.f.: ((log(1-x))^2)/(2*(1-x)^8).

If we define f(n,i,a)=sum(binomial(n,k)*stirling1(n-k,i)*product(-a-j,j=0..k-1),k=0..n-i), then a(n) = |f(n,2,8)|, for n>=1. - Milan Janjic, Dec 21 2008

A249619 Triangle T(m,n) = number of permutations of a multiset with m elements and signature corresponding to n-th integer partition (A194602).

Original entry on oeis.org

1, 1, 2, 1, 6, 3, 1, 24, 12, 4, 6, 1, 120, 60, 20, 30, 5, 10, 1, 720, 360, 120, 180, 30, 60, 6, 90, 15, 20, 1, 5040, 2520, 840, 1260, 210, 420, 42, 630, 105, 140, 7, 210, 21, 35, 1, 40320, 20160, 6720, 10080, 1680, 3360, 336, 5040, 840, 1120, 56

Offset: 0

Tilman Piesk, Nov 04 2014

This triangle shows the same numbers in each row as A036038 and A078760 (the multinomial coefficients), but in this arrangement the multisets in column n correspond to the n-th integer partition in the infinite order defined by A194602.
Row lengths: A000041 (partition numbers), Row sums: A005651
Columns: 0: A000142 (factorials), 1: A001710, 2: A001715, 3: A133799, 4: A001720, 6: A001725, 10: A001730, 14: A049388
Last in row: end-2: A037955 after 1 term mismatch, end-1: A001405, end: A000012
The rightmost columns form the triangle A173333:
  n       0      1     2     4    6  10  14  21      (A000041(1,2,3...)-1)
m
1         1
2         2      1
3         6      3     1
4        24     12     4     1
5       120     60    20     5    1
6       720    360   120    30    6   1
7      5040   2520   840   210   42   7   1
8     40320  20160  6720  1680  336  56   8   1
A249620 shows the number of partitions of the same multisets. A187783 shows the number of permutations of special multisets.

			Triangle begins:
  n     0    1    2    3   4   5  6   7   8   9 10
m
0       1
1       1
2       2    1
3       6    3    1
4      24   12    4    6   1
5     120   60   20   30   5  10  1
6     720  360  120  180  30  60  6  90  15  20  1

Tilman Piesk, Triangle rows m=0..25, flattened
Tilman Piesk, Illustration of the multisets for m=0..6

Cf. A036038, A078760, A005651, A005651, A249620, A000041, A173333.

A373168 Triangle read by rows: the exponential almost-Riordan array ( exp(x/(1-x)) | 1/(1-x), x ).

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 13, 2, 2, 1, 73, 6, 6, 3, 1, 501, 24, 24, 12, 4, 1, 4051, 120, 120, 60, 20, 5, 1, 37633, 720, 720, 360, 120, 30, 6, 1, 394353, 5040, 5040, 2520, 840, 210, 42, 7, 1, 4596553, 40320, 40320, 20160, 6720, 1680, 336, 56, 8, 1, 58941091, 362880, 362880, 181440, 60480, 15120, 3024, 504, 72, 9, 1

Offset: 0

Stefano Spezia, May 26 2024

			The triangle begins:
     1;
     1,   1;
     3,   1,   1;
    13,   2,   2,  1;
    73,   6,   6,  3,  1;
   501,  24,  24, 12,  4, 1;
  4051, 120, 120, 60, 20, 5, 1;
  ...

Y. Alp and E. G. Kocer, Exponential Almost-Riordan Arrays, Results Math 79, 173 (2024). See page 13.

Cf. A000142, A000262 (k=0), A001710, A001715, A001720, A001725, A001730, A049388, A049389, A049398, A051431.

Triangle A094587 with 1st column A000262.

T[n_,0]:=n!SeriesCoefficient[Exp[x/(1-x)],{x,0,n}]; T[n_,k_]:=(n-1)!/(k-1)!SeriesCoefficient[1/(1-x)*x^(k-1),{x,0,n-1}]; Table[T[n,k],{n,0,10},{k,0,n}]//Flatten

T(n,0) = n! * [x^n] exp(x/(1-x)); T(n,k) = (n-1)!/(k-1)! * [x^(n-1)] 1/(1-x)*x^(k-1).
T(n,3) = A001710(n-1) for n > 2.
T(n,4) = A001715(n-1) for n > 3.
T(n,5) = A001720(n-1) for n > 4.
T(n,6) = A001725(n-1) for n > 5.
T(n,7) = A001730(n-1) for n > 6.
T(n,8) = A049388(n-8) for n > 7.
T(n,9) = A049389(n-9) for n > 8.
T(n,10) = A049398(n-10) for n > 9.
T(n,11) = A051431(n-11) for n > 10.

cp's OEIS Frontend

A172455 The case S(6,-4,-1) of the family of self-convolutive recurrences studied by Martin and Kearney.

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A303614 a(n) = [x^n] (1/5040 * Sum_{k=0..n} (k+7)!*x^k)^(1/4).

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A051561 Third unsigned column of triangle A051379.

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A249619 Triangle T(m,n) = number of permutations of a multiset with m elements and signature corresponding to n-th integer partition (A194602).

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A373168 Triangle read by rows: the exponential almost-Riordan array ( exp(x/(1-x)) | 1/(1-x), x ).

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