A053101
a(n) = ((6*n+8)(!^6))/8(!^6), related to A034689 (((6*n+2)(!^6))/2 sextic, or 6-factorials).
Original entry on oeis.org
1, 14, 280, 7280, 232960, 8852480, 389509120, 19475456000, 1090625536000, 67618783232000, 4598077259776000, 340257717223424000, 27220617377873920000, 2340973094497157120000, 215369524693738455040000
Offset: 0
Cf.
A047058,
A008542(n+1),
A034689(n+1),
A034723(n+1),
A034724(n+1),
A034787(n+1),
A034788(n+1),
A053100, this sequence,
A053102,
A053103 (rows m=0..10).
-
m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(1/(1-6*x)^(7/3))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Aug 15 2018
-
s=1;lst={s};Do[s+=n*s;AppendTo[lst, s], {n, 13, 5!, 6}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 08 2008 *)
With[{nn = 30}, CoefficientList[Series[1/(1 - 6*x)^(7/3), {x, 0, nn}], x]*Range[0, nn]!] (* G. C. Greubel, Aug 15 2018 *)
-
x='x+O('x^30); Vec(serlaplace(1/(1-6*x)^(7/3))) \\ G. C. Greubel, Aug 15 2018
A053103
a(n) = ((6*n+10)(!^6))/10(!^6), related to A034724 (((6*n+4)(!^6))/4 sextic, or 6-factorials).
Original entry on oeis.org
1, 16, 352, 9856, 335104, 13404160, 616591360, 32062750720, 1859639541760, 119016930672640, 8331185147084800, 633170071178444800, 51919945836632473600, 4568955233623657676800, 429481791960623821619200
Offset: 0
Cf.
A047058,
A008542(n+1),
A034689(n+1),
A034723(n+1),
A034724(n+1),
A034787(n+1),
A034788(n+1),
A053100,
A053101,
A053102, this sequence (rows m=0..10).
-
m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(1/(1-6*x)^(8/3))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Aug 16 2018
-
s=1;lst={s};Do[s+=n*s;AppendTo[lst, s], {n, 15, 5!, 6}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 08 2008 *)
With[{nn = 30}, CoefficientList[Series[1/(1 - 6*x)^(16/6), {x, 0, nn}], x]*Range[0, nn]!] (* G. C. Greubel, Aug 16 2018 *)
-
x='x+O('x^30); Vec(serlaplace(1/(1-6*x)^(8/3))) \\ G. C. Greubel, Aug 16 2018
A053102
a(n) = ((6*n+9)(!^6))/9(!^6), related to A034723 (((6*n+3)(!^6))/3 sextic, or 6-factorials).
Original entry on oeis.org
1, 15, 315, 8505, 280665, 10945935, 492567075, 25120920825, 1431892487025, 90209226682575, 6224436641097675, 466832748082325625, 37813452594668375625, 3289770375736148679375, 305948644943461827181875
Offset: 0
Cf.
A047058,
A008542(n+1),
A034689(n+1),
A034723(n+1),
A034724(n+1),
A034787(n+1),
A034788(n+1),
A053100,
A053101, this sequence,
A053103 (rows m=0..10).
-
m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(1/(1-6*x)^(15/6))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Aug 15 2018
-
s=1;lst={s};Do[s+=n*s;AppendTo[lst, s], {n, 14, 5!, 6}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 08 2008 *)
With[{nn = 30}, CoefficientList[Series[1/(1 - 6*x)^(15/6), {x, 0, nn}], x]*Range[0, nn]!] (* G. C. Greubel, Aug 15 2018 *)
-
x='x+O('x^30); Vec(serlaplace(1/(1-6*x)^(15/6))) \\ G. C. Greubel, Aug 15 2018
A172455
The case S(6,-4,-1) of the family of self-convolutive recurrences studied by Martin and Kearney.
Original entry on oeis.org
1, 7, 84, 1463, 33936, 990542, 34938624, 1445713003, 68639375616, 3676366634402, 219208706540544, 14397191399702118, 1032543050697424896, 80280469685284582812, 6725557192852592984064, 603931579625379293509683
Offset: 1
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).
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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 *)
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{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
Showing 1-4 of 4 results.
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