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A274964 G.f.: 1 = ...(((1/(1-x) - a(1)*0!*x )^2 - a(2)*1!*x^2 )^3 - a(3)*2!*x^3 )^4 - a(4)*3!*x^4 )^5 -..., an infinite series of nested powers. 2
1, 2, 3, 6, 10, 27, 63, 290, 732, 3950, 14773, 119427, 492167, 4739742, 29304550, 503805954, 2394731095, 33364109691, 267779096008, 6778552047060, 47933808592698, 1141492809790630, 13812159440293116, 631536544942731743, 5059663373178085810, 161991873932527358590, 2458252827155843942412, 157277460804140189938404, 1910359108470010215694334, 104035730915390562574302557, 2638816703576929939008316379, 345342948094489081419419253954 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

LINKS

Paul D. Hanna, Table of n, a(n) for n = 1..300

EXAMPLE

G.f.: 1 = ... (((((((((1/(1-x) - 1*0!*x)^2 - 2*1!*x^2)^3 - 3*2!*x^3)^4 - 6*3!*x^4)^5 - 10*4!*x^5)^6 - 27*5!*x^6)^7 - 63*6!*x^7)^8 - 290*7!*x^8)^9 - 732*8!*x^9)^10 -...- a(n)*(n-1)!*x^n )^(n+1) -...

ILLUSTRATION OF GENERATING METHOD.

Start with G1 = 1/(1-x), and proceed as follows:

G2 = (G1 - 1*0!*x)^2 = 1 + 2*x^2 + 2*x^3 + 3*x^4 + 4*x^5 + 5*x^6 + 6*x^7 + 7*x^8 +...

G3 = (G2 - 2*1!*x^2)^3 = 1 + 6*x^3 + 9*x^4 + 12*x^5 + 27*x^6 + 54*x^7 + 96*x^8 +...

G4 = (G3 - 3*2!*x^3)^4 = 1 + 36*x^4 + 48*x^5 + 108*x^6 + 216*x^7 + 870*x^8 +...

G5 = (G4 - 6*3!*x^4)^5 = 1 + 240*x^5 + 540*x^6 + 1080*x^7 + 4350*x^8 + 9760*x^9 +...

G6 = (G5 - 10*4!*x^5)^6 = 1 + 3240*x^6 + 6480*x^7 + 26100*x^8 + 58560*x^9 +...

G7 = (G6 - 27*5!*x^6)^7 = 1 + 45360*x^7 + 182700*x^8 + 409920*x^9 + 1990800*x^10 +...

G8 = (G7 - 63*6!*x^7)^8 = 1 + 1461600*x^8 + 3279360*x^9 + 15926400*x^10 +...

G9 = (G8 - 290*7!*x^8)^9 = 1 + 29514240*x^9 + 143337600*x^10 + 487347840*x^11 +...

G10 = (G9 - 732*8!*x^9)^10 = 1 + 1433376000*x^10 + 4873478400*x^11 + 36114724800*x^12 +...

...

G_{n+1} = (G_{n} - a(n)*(n-1)!*x^n)^(n+1) = 1 + a(n+1)*n!*x^(n+1) +...

...

PROG

(PARI) {a(n) = my(A=[1], G = 1/(1-x +x^2*O(x^n)));

for(m=1, n-1, A=concat(A, 0); G = (G - A[m]*(m-1)!*x^m)^(m+1);

A[m+1] = polcoeff(G, m+1)/m!; ); A[n]}

for(n=1, 30, print1(a(n), ", "))

(PARI) /* Informal quick print of the initial N terms: */

{N=100; A=[1]; G = 1/(1-x +x^2*O(x^N));

for(m=1, N-1, A=concat(A, 0); G = (G - A[m]*(m-1)!*x^m)^(m+1);

A[m+1] = polcoeff(G, m+1)/m!; print1(A[m], ", "); ); print1(A[N], ", ")}

CROSSREFS

Cf. A274960.

Sequence in context: A212606 A192440 A327711 * A054357 A056606 A186408

Adjacent sequences:  A274961 A274962 A274963 * A274965 A274966 A274967

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jul 12 2016

STATUS

approved

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Last modified August 13 06:22 EDT 2020. Contains 336442 sequences. (Running on oeis4.)