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 A100075 Triangle, read by rows, of coefficients in powers of e.g.f. for A100076 such that, for each row n>=0, Sum_{k=0..n} T(n,k)/k! = [sqrt(5)^n]. 1
 1, 1, 1, 1, 2, 4, 1, 3, 9, 15, 1, 4, 16, 48, 96, 1, 5, 25, 105, 345, 555, 1, 6, 36, 192, 864, 2772, 6408, 1, 7, 49, 315, 1785, 8169, 28665, 59157, 1, 8, 64, 480, 3264, 19056, 91296, 323424, 734976, 1, 9, 81, 693, 5481, 38583, 233361, 1144611, 4222449, 9129591, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 LINKS EXAMPLE Rows form the initial coefficients of powers of e.g.f. of A100076: G100076^0: [1,__0,0,0,0,0,0,0,0,...], G100076^1: [1,1,__1,-3,9,-33,513,-10917,155313,...], G100076^2: [1,2,4,__0,0,-36,1080,-17928,181440,...], G100076^3: [1,3,9,15,__9,-99,1521,-17631,112401,...], G100076^4: [1,4,16,48,96,__-72,1296,-11664,31104,...], G100076^5: [1,5,25,105,345,555,__1305,-6705,-6255,...], G100076^6: [1,6,36,192,864,2772,6408,__648,-15552,...], G100076^7: [1,7,49,315,1785,8169,28665,59157,__41265,...],... such that for each row n, Sum_{k=0..n} T(n,k)/k! = [sqrt(5)^n]: [sqrt(5)^0] = 1 = 1 [sqrt(5)^1] = 1+1 = 2 [sqrt(5)^2] = 1+2+4/2! = 5 [sqrt(5)^3] = 1+3+9/2!+15/3! = 11 [sqrt(5)^4] = 1+4+16/2!+48/3!+96/4! = 25 [sqrt(5)^5] = 1+5+25/2!+105/3!+345/4!+555/5! = 55 [sqrt(5)^6] = 1+6+36/2!+192/3!+864/4!+2772/5!+6408/6! = 125 [sqrt(5)^7] = 1+7+49/2!+315/3!+1785/4!+8169/5!+28665/6!+59157/7! = 279 PROG (PARI) {T(n, k)=if(n==0, 1, if(k==0, 1, if(k==n, n!*(floor(sqrt(5)^n+1/10^15)-sum(j=0, n-1, T(n, j)/j!)), k!*polcoeff((Ser(vector(n, i, T(n-1, i-1)/(i-1)!), x)+x*O(x^k))^(n/(n-1)), k, x))))} CROSSREFS Cf. A100076, A100064. Sequence in context: A275895 A158613 A209573 * A059836 A069270 A079901 Adjacent sequences:  A100072 A100073 A100074 * A100076 A100077 A100078 KEYWORD nonn,tabl AUTHOR Paul D. Hanna, Nov 03 2004 STATUS approved

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Last modified August 11 09:52 EDT 2020. Contains 336423 sequences. (Running on oeis4.)