This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A365095 #11 Sep 04 2023 06:05:06 %S A365095 1,1,4,27,256,3118,46114,797049,15671350,343712542,8287906284, %T A365095 217309849772,6143454613682,186012988954448,5999891924386246, %U A365095 205262374717093101,7420869162700453174,282640364822610119566,11310634300879858185320,474456517209788353301282,20818983374432724237753352 %N A365095 Expansion of g.f. A(x) satisfying [x^(n-1)] (1 + (n-1)*x*A(x)^2)^n / A(x)^n = 0 for n > 1. %C A365095 Related identities for the Catalan function C(x) = 1 + x*C(x)^2 (cf. A000108): %C A365095 (1) [x^(n-1)] (1 + (n-1)*x*C(x))^n / C(x)^n = 0 for n > 1. %C A365095 (2) [x^(n-1)] (1 + n*x*C(x)^2)^n / C(x)^(2*n) = 0 for n > 1. %C A365095 (3) [x^(n-1)] (1 + n*x*C(x))^n / C(x)^n = n^(n-1) for n >= 1. %C A365095 (4) [x^(n-1)] (1 + (n+1)*x*C(x)^2)^n / C(x)^(2*n) = n^(n-1) for n >= 1. %H A365095 Paul D. Hanna, <a href="/A365095/b365095.txt">Table of n, a(n) for n = 0..400</a> %F A365095 G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas. %F A365095 (1) [x^(n-1)] (1 + (n-1)*x*A(x)^2)^n / A(x)^n = 0. %F A365095 (2) [x^(n-1)] (1 + (k*n-1)*x*A(x)^2)^n / A(x)^n is divisible by n^2 for n > 0 and all integer k (conjecture). %e A365095 G.f.: A(x) = 1 + x + 4*x^2 + 27*x^3 + 256*x^4 + 3118*x^5 + 46114*x^6 + 797049*x^7 + 15671350*x^8 + 343712542*x^9 + 8287906284*x^10 + ... %e A365095 RELATED SERIES. %e A365095 (1) The power series B(x) = A(x/B(x)) where A(x) = B(x*A(x)) begins %e A365095 B(x) = 1 + x + 3*x^2 + 17*x^3 + 151*x^4 + 1817*x^5 + 27041*x^6 + 472297*x^7 + 9377293*x^8 + 207254037*x^9 + 5025044843*x^10 + ... %e A365095 and appears to have only odd coefficients. %e A365095 (2) The power series D(x) = A(x/D(x)^2) where A(x) = D(x*A(x)^2) begins %e A365095 D(x) = 1 + x + 2*x^2 + 10*x^3 + 90*x^4 + 1106*x^5 + 16684*x^6 + 293796*x^7 + 5860280*x^8 + 129807560*x^9 + 3149052896*x^10 + ... %e A365095 modulo 16 of which appears to equal the series %e A365095 D(x) (mod 16) = (1 + x + 2*x^2 + 10*x^3 + 10*x^4 + 2*x^5 + 12*x^6 + 4*x^7 + 7*x^8 + 7*x^9 - 2*x^10 - 2*x^11 - 10*x^12 + 6*x^13 - 12*x^14 + 4*x^15 - 8*x^16 - 8*x^17) / (1 - x^8). %e A365095 Explicitly, %e A365095 D(x) (mod 16) = 1 + x + 2*x^2 + 10*x^3 + 10*x^4 + 2*x^5 + 12*x^6 + 4*x^7 + 8*x^8 + 8*x^9 + 8*x^11 + 8*x^13 + 8*x^15 + 8*x^19 + 8*x^21 + 8*x^23 + ... %e A365095 RELATED TABLES. %e A365095 The table of coefficients of x^k in (1 + (n-1)*x*A(x)^2)^n/A(x)^n begins: %e A365095 n=1: [1, -1, -3, -20, -197, -2504, -38396, ...]; %e A365095 n=2: [1, 0, -4, -32, -336, -4432, -69620, ...]; %e A365095 n=3: [1, 3, 0, -41, -501, -6795, -107500, ...]; %e A365095 n=4: [1, 8, 24, 0, -640, -10112, -163272, ...]; %e A365095 n=5: [1, 15, 95, 310, 0, -13027, -246265, ...]; %e A365095 n=6: [1, 24, 252, 1520, 5448, 0, -321580, ...]; %e A365095 n=7: [1, 35, 546, 5033, 30534, 119728, 0, ...]; ... %e A365095 in which the main diagonal equals all zeros after the initial term, illustrating that [x^(n-1)] (1 + (n-1)*x*A(x)^2)^n / A(x)^n = 0 for n > 1. %e A365095 The table of coefficients of x^k in A(x)^n begins: %e A365095 n=1: [1, 1, 4, 27, 256, 3118, 46114, ...]; %e A365095 n=2: [1, 2, 9, 62, 582, 6964, 101241, ...]; %e A365095 n=3: [1, 3, 15, 106, 990, 11667, 166861, ...]; %e A365095 n=4: [1, 4, 22, 160, 1493, 17372, 244658, ...]; %e A365095 n=5: [1, 5, 30, 225, 2105, 24241, 336540, ...]; %e A365095 n=6: [1, 6, 39, 302, 2841, 32454, 444660, ...]; %e A365095 n=7: [1, 7, 49, 392, 3717, 42210, 571438, ...]; ... %o A365095 (PARI) {a(n) = my(A=[1]); for(i=1,n, A=concat(A,0); m=#A; %o A365095 A[#A] = polcoeff( (1 + (m-1)*x*Ser(A)^2)^m / Ser(A)^m , m-1)/m ); A[n+1]} %o A365095 for(n=0,30,print1(a(n),", ")) %Y A365095 Cf. A303063. %K A365095 nonn %O A365095 0,3 %A A365095 _Paul D. Hanna_, Sep 03 2023