The OEIS is supported by the many generous donors to the OEIS Foundation.

 Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 60th year, we have over 367,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”). Other ways to Give
 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A251665 E.g.f.: exp(5*x*G(x)^4) / G(x) where G(x) = 1 + x*G(x)^5 is the g.f. of A002294. 10
 1, 4, 47, 1034, 34349, 1540480, 87311275, 5991370390, 483100288985, 44778459212540, 4691799973171175, 548418557098305250, 70754785462138421125, 9987462340422594014200, 1531136319790275407365475, 253347224928445454055920750, 45001449932636667231257800625, 8541130421294458307989700672500 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 LINKS Table of n, a(n) for n=0..17. FORMULA Let G(x) = 1 + x*G(x)^5 be the g.f. of A002294, then the e.g.f. A(x) of this sequence satisfies: (1) A'(x)/A(x) = G(x)^4 + 3*G'(x)/G(x). (2) A(x) = F(x/A(x)^4) where F(x) is the e.g.f. of A251695. (3) A(x) = Sum_{n>=0} A251695(n)*(x/A(x)^4)^n/n! where A251695(n) = (3*n+1) * (4*n+1)^(n-2) * 5^n . (4) [x^n/n!] A(x)^(4*n+1) = (3*n+1) * (4*n+1)^(n-1) * 5^n. a(n) = Sum_{k=0..n} 5^k * n!/k! * binomial(5*n-k-2,n-k) * (4*k-1)/(4*n-1) for n>=0. Recurrence: 8*(2*n-1)*(4*n-3)*(4*n-1)*(1875*n^4 - 13375*n^3 + 35700*n^2 - 41905*n + 17681)*a(n) = 5*(1171875*n^8 - 11875000*n^7 + 51765625*n^6 - 126596875*n^5 + 189126875*n^4 - 174442875*n^3 + 93137550*n^2 - 22362645*n - 233856)*a(n-1) - 3125*(1875*n^4 - 5875*n^3 + 6825*n^2 - 3130*n - 24)*a(n-2). - Vaclav Kotesovec, Dec 07 2014 a(n) ~ 3 * 5^(5*n-3/2) / 2^(8*n-1) * n^(n-1) / exp(n-1). - Vaclav Kotesovec, Dec 07 2014 EXAMPLE E.g.f.: A(x) = 1 + 4*x + 47*x^2/2! + 1034*x^3/3! + 34349*x^4/4! + 1540480*x^5/5! +... such that A(x) = exp(5*x*G(x)^4) / G(x) where G(x) = 1 + x*G(x)^5 is the g.f. of A002294: G(x) = 1 + x + 5*x^2 + 35*x^3 + 285*x^4 + 2530*x^5 + 23751*x^6 +... MATHEMATICA Table[Sum[5^k * n!/k! * Binomial[5*n-k-2, n-k] * (4*k-1)/(4*n-1), {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Dec 07 2014 *) PROG (PARI) {a(n)=local(G=1); for(i=0, n, G=1+x*G^5 +x*O(x^n)); n!*polcoeff(exp(5*x*G^4)/G, n)} for(n=0, 20, print1(a(n), ", ")) (PARI) {a(n) = sum(k=0, n, 5^k * n!/k! * binomial(5*n-k-2, n-k) * (4*k-1)/(4*n-1) )} for(n=0, 20, print1(a(n), ", ")) CROSSREFS Cf. A251575, A251695, A002294. Cf. Variants: A243953, A251663, A251664, A251666, A251667, A251668, A251669, A251670. Sequence in context: A326434 A319833 A006438 * A210828 A361560 A333247 Adjacent sequences: A251662 A251663 A251664 * A251666 A251667 A251668 KEYWORD nonn AUTHOR Paul D. Hanna, Dec 07 2014 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified December 7 08:27 EST 2023. Contains 367645 sequences. (Running on oeis4.)