A051188
Sept-factorial numbers.
Original entry on oeis.org
1, 7, 98, 2058, 57624, 2016840, 84707280, 4150656720, 232436776320, 14643516908160, 1025046183571200, 78928556134982400, 6629998715338521600, 603329883095805465600, 59126328543388935628800
Offset: 0
Cf.
A000142,
A000165,
A032031,
A045754,
A047053,
A047058,
A049209,
A051186,
A052562,
A053106,
A084947,
A092516,
A092750,
A144827,
A144739,
A147585.
A053104
a(n) = ((7*n+8)(!^7))/8, related to A045754 ((7*n+1)(!^7) sept-, or 7-factorials).
Original entry on oeis.org
1, 15, 330, 9570, 344520, 14814360, 740718000, 42220926000, 2702139264000, 191851887744000, 14964447244032000, 1271978015742720000, 117021977448330240000, 11585175767384693760000
Offset: 0
-
m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(1/(1-7*x)^(15/7))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Aug 16 2018
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s=1;lst={s};Do[s+=n*s;AppendTo[lst, s], {n, 14, 5!, 7}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 08 2008 *)
With[{nn = 30}, CoefficientList[Series[1/(1 - 7*x)^(15/7), {x, 0, nn}], x]*Range[0, nn]!] (* G. C. Greubel, Aug 16 2018 *)
-
x='x+O('x^30); Vec(serlaplace(1/(1-7*x)^(15/7))) \\ G. C. Greubel, Aug 16 2018
A053105
a(n) = ((7*n+9)(!^7))/9(!^7), related to A034829 (((7*n+2)(!^7))/2 sept-, or 7-factorials).
Original entry on oeis.org
1, 16, 368, 11040, 408480, 17973120, 916629120, 53164488960, 3455691782400, 248809808332800, 19655974858291200, 1690413837813043200, 157208486916613017600, 15720848691661301760000
Offset: 0
-
m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(1/(1-7*x)^(16/7))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Aug 16 2018
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s=1;lst={s};Do[s+=n*s;AppendTo[lst, s], {n, 15, 5!, 7}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 08 2008 *)
CoefficientList[Series[1/(1-7x)^(16/7),{x,0,20}],x]Range[0,20]! (* Harvey P. Dale, Sep 11 2011 *)
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x='x+O('x^30); Vec(serlaplace(1/(1-7*x)^(16/7))) \\ G. C. Greubel, Aug 16 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 */
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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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