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 A352415 G.f. A(x) satisfies: A(x)^6 = (1-x) * (A(x) + x)^5. 2
 1, 4, -15, 95, -815, 7881, -81946, 894100, -10097235, 117019845, -1383816039, 16631112009, -202545350505, 2494192904025, -31003553499195, 388498706407341, -4902323847971661, 62240419152427905, -794494411812382465, 10190561785036460125 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 LINKS Table of n, a(n) for n=0..19. FORMULA G.f. A(x) satisfies: (1) A(x) = ( A(x)^6/(1-x) )^(1/5) - x. (2) A(x)^6 = (1-x) * (A(x) + x)^5. (3) A( x*(1+x)^5/(1 + x*(1+x)^5) ) = (1+x)^5/(1 + x*(1+x)^5). (4) A(x) = x / Series_Reversion( x*(1+x)^5/(1 + x*(1+x)^5) ). (5) Sum_{k=0..n} [x^k] A(x)^n = (-1)^(n-1) * 5, for n >= 1. EXAMPLE G.f.: A(x) = 1 + 4*x - 15*x^2 + 95*x^3 - 815*x^4 + 7881*x^5 - 81946*x^6 + 894100*x^7 - 10097235*x^8 + ... where A(x)^6 equals (1-x)*(A(x) + x)^5, as can be seen from the following power series expansions: A(x)^6 = 1 + 24*x + 150*x^2 + 50*x^3 - 675*x^4 + 480*x^5 - 35*x^6 + 1980*x^7 + ... (A(x) + x)^5 = 1 + 25*x + 175*x^2 + 225*x^3 - 450*x^4 + 30*x^5 - 5*x^6 + 1975*x^7 + ... Related table. Another defining property of the g.f. A(x) is illustrated here. The table of coefficients of x^k in A(x)^n begins: n=1: [1, 4, -15, 95, -815, 7881, -81946, 894100, ...]; n=2: [1, 8, -14, 70, -645, 6392, -67369, 741352, ...]; n=3: [1, 12, 3, -11, -210, 2793, -32964, 385869, ...]; n=4: [1, 16, 36, -84, 26, 504, -9506, 135524, ...]; n=5: [1, 20, 85, -85, -145, 129, -1050, 27550, ...]; n=6: [1, 24, 150, 50, -675, 480, -35, 1980, ...]; n=7: [1, 28, 231, 385, -1260, -399, 1708, -689, ...]; ... in which the partial sum of row n up to column n equals (-1)^(n-1)*5, as illustrated by: n=1: 5 = 1 + 4; n=2: -5 = 1 + 8 + -14; n=3: 5 = 1 + 12 + 3 + -11; n=4: -5 = 1 + 16 + 36 + -84 + 26; n=5: 5 = 1 + 20 + 85 + -85 + -145 + 129; n=6: -5 = 1 + 24 + 150 + 50 + -675 + 480 + -35; n=7: 5 = 1 + 28 + 231 + 385 + -1260 + -399 + 1708 + -689; ... PROG (PARI) {a(n) = polcoeff( x/serreverse( x*(1+x)^5/(1 + x*(1+x)^5 +x^2*O(x^n)) ), n)} for(n=0, 30, print1(a(n), ", ")) CROSSREFS Cf. A352385, A352413, A352414. Sequence in context: A366697 A208991 A109365 * A079128 A356524 A289489 Adjacent sequences: A352412 A352413 A352414 * A352416 A352417 A352418 KEYWORD sign AUTHOR Paul D. Hanna, Mar 15 2022 STATUS approved

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Last modified August 3 05:44 EDT 2024. Contains 374875 sequences. (Running on oeis4.)