cp's OEIS Frontend

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.

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.

Original entry on oeis.org

1, 1, 4, 27, 256, 3118, 46114, 797049, 15671350, 343712542, 8287906284, 217309849772, 6143454613682, 186012988954448, 5999891924386246, 205262374717093101, 7420869162700453174, 282640364822610119566, 11310634300879858185320, 474456517209788353301282, 20818983374432724237753352
Offset: 0

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Author

Paul D. Hanna, Sep 03 2023

Keywords

Comments

Related identities for the Catalan function C(x) = 1 + x*C(x)^2 (cf. A000108):
(1) [x^(n-1)] (1 + (n-1)*x*C(x))^n / C(x)^n = 0 for n > 1.
(2) [x^(n-1)] (1 + n*x*C(x)^2)^n / C(x)^(2*n) = 0 for n > 1.
(3) [x^(n-1)] (1 + n*x*C(x))^n / C(x)^n = n^(n-1) for n >= 1.
(4) [x^(n-1)] (1 + (n+1)*x*C(x)^2)^n / C(x)^(2*n) = n^(n-1) for n >= 1.

Examples

			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 + ...
RELATED SERIES.
(1) The power series B(x) = A(x/B(x)) where A(x) = B(x*A(x)) begins
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 + ...
and appears to have only odd coefficients.
(2) The power series D(x) = A(x/D(x)^2) where A(x) = D(x*A(x)^2) begins
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 + ...
modulo 16 of which appears to equal the series
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).
Explicitly,
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 + ...
RELATED TABLES.
The table of coefficients of x^k in (1 + (n-1)*x*A(x)^2)^n/A(x)^n begins:
n=1: [1, -1,   -3,   -20,  -197,  -2504,  -38396, ...];
n=2: [1,  0,   -4,   -32,  -336,  -4432,  -69620, ...];
n=3: [1,  3,    0,   -41,  -501,  -6795, -107500, ...];
n=4: [1,  8,   24,     0,  -640, -10112, -163272, ...];
n=5: [1, 15,   95,   310,     0, -13027, -246265, ...];
n=6: [1, 24,  252,  1520,  5448,      0, -321580, ...];
n=7: [1, 35,  546,  5033, 30534, 119728,       0, ...]; ...
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.
The table of coefficients of x^k in A(x)^n begins:
n=1: [1, 1,  4,  27,  256,  3118,  46114, ...];
n=2: [1, 2,  9,  62,  582,  6964, 101241, ...];
n=3: [1, 3, 15, 106,  990, 11667, 166861, ...];
n=4: [1, 4, 22, 160, 1493, 17372, 244658, ...];
n=5: [1, 5, 30, 225, 2105, 24241, 336540, ...];
n=6: [1, 6, 39, 302, 2841, 32454, 444660, ...];
n=7: [1, 7, 49, 392, 3717, 42210, 571438, ...]; ...
		

Crossrefs

Cf. A303063.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=1,n, A=concat(A,0); m=#A;
    A[#A] = polcoeff( (1 + (m-1)*x*Ser(A)^2)^m / Ser(A)^m , m-1)/m ); A[n+1]}
    for(n=0,30,print1(a(n),", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) [x^(n-1)] (1 + (n-1)*x*A(x)^2)^n / A(x)^n = 0.
(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).