This site is supported by donations to The OEIS Foundation.

 Please make a donation to keep the OEIS running. We are now in our 55th year. In the past year we added 12000 new sequences and reached 8000 citations (which often say "discovered thanks to the OEIS"). We need to raise money to hire someone to manage submissions, which would reduce the load on our editors and speed up editing. Other ways to donate

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A326005 G.f.: Sum_{n>=0} (n+1)*(n+2)*(n+3)*(n+4)/4! * x^n * (1 + x^n)^n. 3
 1, 5, 20, 35, 100, 126, 330, 330, 775, 820, 1631, 1365, 3535, 2380, 5370, 5136, 9085, 5985, 16900, 8855, 21966, 19580, 29965, 17550, 60375, 24381, 58345, 57205, 90350, 40920, 152837, 52360, 164145, 141120, 175560, 93801, 404500, 101270, 280175, 309050, 503041, 148995, 714435, 178365, 748705, 708946, 633950, 249900, 1771645, 295135, 1120236, 1155015, 1760500, 395010, 2483110, 905576, 2622545, 2036060, 1744525, 595665, 6962328, 677040, 2343880 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS More generally, the following sums are equal: (1) Sum_{n>=0} binomial(n+k-1, n) * r^n * (p + q^n)^n, (2) Sum_{n>=0} binomial(n+k-1, n) * r^n * q^(n^2) / (1 - p*q^n*r)^(n+k), for any fixed integer k; here, k = 5 and p = 1, q = x, r = x. LINKS FORMULA Generating functions. (1) Sum_{n>=0} (n+1)*(n+2)*(n+3)*(n+4)/4! * x^n * (1 + x^n)^n. (2) Sum_{n>=0} (n+1)*(n+2)*(n+3)*(n+4)/4! * x^(n*(n+1)) / (1 - x^(n+1))^(n+5). EXAMPLE G.f.: A(x) = 1 + 5*x + 20*x^2 + 35*x^3 + 100*x^4 + 126*x^5 + 330*x^6 + 330*x^7 + 775*x^8 + 820*x^9 + 1631*x^10 + 1365*x^11 + 3535*x^12 + 2380*x^13 + 5370*x^14 + 5136*x^15 + 9085*x^16 + 5985*x^17 + 16900*x^18 + 8855*x^19 + 21966*x^20 + ... where we have the following series identity: A(x) = 1 + 5*x*(1+x) + 15*x^2*(1+x^2)^2 + 35*x^3*(1+x^3)^3 + 70*x^4*(1+x^4)^4 + 126*x^5*(1+x^5)^5  + 210*x^6*(1+x^6)^6 + 330*x^7*(1+x^7)^7 + 495*x^8*(1+x^8)^8 + 715*x^9*(1+x^9)^9 +... is equal to A(x) = 1/(1-x)^5 + 5*x^2/(1-x^2)^6 + 15*x^6/(1-x^3)^7 + 35*x^12/(1-x^4)^8 + 70*x^20/(1-x^5)^9 + 126*x^30/(1-x^6)^10 + 210*x^42/(1-x^7)^11 + 330*x^56/(1-x^8)^12 +... PROG (PARI) {a(n) = my(A = sum(m=0, n, (m+1)*(m+2)*(m+3)*(m+4)/4! * x^m * (1 + x^m +x*O(x^n))^m)); polcoeff(A, n)} for(n=0, 120, print1(a(n), ", ")) (PARI) {a(n) = my(A = sum(m=0, n, (m+1)*(m+2)*(m+3)*(m+4)/4! * x^m * x^(m^2) / (1 - x^(m+1) +x*O(x^n))^(m+5))); polcoeff(A, n)} for(n=0, 120, print1(a(n), ", ")) CROSSREFS Cf. A217668 (k=1), A326002 (k=2), A326003 (k=3), A326004 (k=4). Sequence in context: A101867 A003339 A047716 * A063110 A044066 A013337 Adjacent sequences:  A326002 A326003 A326004 * A326006 A326007 A326008 KEYWORD nonn AUTHOR Paul D. Hanna, Jun 01 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 | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified December 13 22:55 EST 2019. Contains 329974 sequences. (Running on oeis4.)