The OEIS mourns the passing of Jim Simons and is grateful to the Simons Foundation for its support of research in many branches of science, including the OEIS.
The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A326561 G.f. A(x) satisfies: Sum_{n>=0} A(x)^(n*(n+1)+1) * x^n = Sum_{n>=0} (A(x)^(n+1) + 1)^n * x^n. 7
 1, 1, 2, 8, 43, 268, 1831, 13354, 102352, 816241, 6728037, 57056328, 496185294, 4414563737, 40114493041, 371845437684, 3513262944971, 33815076029376, 331454445914861, 3308183361941640, 33620505978224843, 347942136527114740, 3667458554727506170, 39378668472879902163, 430803711668467138362, 4802830669726993050928, 54572510428547279296599 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 LINKS Paul D. Hanna, Table of n, a(n) for n = 0..250 FORMULA G.f. A(x) allows the following sums to be equal: (1) B(x) = Sum_{n>=0} A(x)^(n*(n+1)+1) * x^n. (2) B(x) = Sum_{n>=0} (A(x)^(n+1) + 1)^n * x^n. (3) B(x) = Sum_{n>=0} A(x)^(n*(n+1)) * x^n / (1 - x*A(x)^n)^(n+1). EXAMPLE G.f.: A(x) = 1 + x + 2*x^2 + 8*x^3 + 43*x^4 + 268*x^5 + 1831*x^6 + 13354*x^7 + 102352*x^8 + 816241*x^9 + 6728037*x^10 + 57056328*x^11 + 496185294*x^12 + ... such that the following sums are equal B(x) = A(x) + A(x)^3*x + A(x)^7*x^2 + A(x)^13*x^3 + A(x)^21*x^4 + A(x)^31*x^5 + A(x)^43*x^6 + A(x)^57*x^7 + A(x)^73*x^8 + ... + A(x)^(n*(n+1)+1)*x^n + ... and B(x) = 1 + (1 + A(x)^2)*x + (1 + A(x)^3)^2*x^2 + (1 + A(x)^4)^3*x^3 + (1 + A(x)^5)^4*x^4 + (1 + A(x)^6)^5*x^5 + ... + (1 + A(x)^(n+1))^n*x^n + ... also B(x) = 1/(1 - x) + A(x)^2*x/(1 - x*A(x))^2 + A(x)^6*x^2/(1 - x*A(x)^2)^3 + A(x)^12*x^3/(1 - x*A(x)^3)^4 + ... + A(x)^(n*(n+1))*x^n/(1 - x*A(x)^n)^(n+1) + ... where B(x) = 1 + 2*x + 6*x^2 + 25*x^3 + 129*x^4 + 764*x^5 + 4977*x^6 + 34770*x^7 + 256358*x^8 + 1973671*x^9 + 15750935*x^10 + 129624972*x^11 + 1095963211*x^12 + ... PROG (PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); A[#A] = polcoeff( sum(m=0, #A, (Ser(A)^(m+1) + 1)^m*x^m - Ser(A)^(m^2+m+1)*x^m ), #A-1)); A[n+1]} for(n=0, 30, print1(a(n), ", ")) CROSSREFS Cf. A326560, A326562, A326563, A326275, A326287. Sequence in context: A193659 A020023 A027836 * A007169 A106327 A009309 Adjacent sequences: A326558 A326559 A326560 * A326562 A326563 A326564 KEYWORD nonn AUTHOR Paul D. Hanna, Jul 12 2019 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 June 17 09:14 EDT 2024. Contains 373444 sequences. (Running on oeis4.)