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A265942 G.f. A(x) satisfies: x = Sum_{n>=1} x^n * A(-n*x). 0
1, 1, 1, 2, 10, 118, 3074, 170402, 19685482, 4679048902, 2269261320050, 2232902808762146, 4440206282955006346, 17793903288234980266774, 143417291429469295762696226, 2321454058916677584825960932258, 75384639189021086218383176211366826, 4907135862070128733441621809087271706086, 639945541949598669937210207492354528460950418, 167124499527470730841323059848701978687632400092450 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

LINKS

Table of n, a(n) for n=0..19.

FORMULA

a(n) = Sum_{k=0..n-1} (-1)^(n-k+1) * (n-k+1)^k * a(k), for n>0 with a(0)=1.

0 = Sum_{k=0..n} (-1)^k * (n-k)^k * a(k), for n>1.

a(n) = 2 (mod 4) for n>2 [conjecture].

EXAMPLE

G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 10*x^4 + 118*x^5 + 3074*x^6 + 170402*x^7 + 19685482*x^8 + 4679048902*x^9 + 2269261320050*x^10 +...

where

x = x*A(-x) + x^2*A(-2*x) + x^3*A(-3*x) + x^4*A(-4*x) + x^5*A(-5*x) + x^6*A(-6*x) + x^7*A(-7*x) + x^8*A(-8*x) + x^9*A(-9*x) +...

The array of coefficients in A(-n*x) begins:

n=1: [1, -1, 1, -2, 10, -118, 3074, -170402, 19685482]

n=2: [1, -2, 4, -16, 160, -3776, 196736, -21811456, ...];

n=3: [1, -3, 9, -54, 810, -28674, 2240946, -372669174, ...];

n=4: [1, -4, 16, -128, 2560, -120832, 12591104, -2791866368, ...];

n=5: [1, -5, 25, -250, 6250, -368750, 48031250, -13312656250, ...];

n=6: [1, -6, 36, -432, 12960, -917568, 143420544, -47701654272, ...];

n=7: [1, -7, 49, -686, 24010, -1983226, 361653026, -140333374286, ...];

n=8: [1, -8, 64, -1024, 40960, -3866624, 805830656, -357358895104, ...]; ...

in which the antidiagonal sums yield [1,0,0,0,0,0,0,0,...].

PROG

(PARI) {a(n)=my(A=[1]); for(i=1, n, A=concat(A, 0); A[#A]=(-1)^(#A)*Vec(sum(m=1, #A, subst(Ser(A), x, -m*x)*x^m))[#A]); A[n+1]}

for(n=0, 30, print1(a(n), ", "))

(PARI) {a(n) = if(n==0, 1, sum(k=0, n-1, (-1)^(n-k+1)*a(k)*(n-k+1)^k ) )}

for(n=0, 20, print1(a(n), ", "))

(PARI) /* Quick print of terms 0..30 */

{A=[1]; for(i=1, 30, A=concat(A, 0);

A[#A]=(-1)^(#A)*Vec(sum(n=1, #A, subst(Ser(A), x, -n*x)*x^n))[#A] ); A}

CROSSREFS

Sequence in context: A006121 A110951 A172477 * A120597 A322295 A256832

Adjacent sequences:  A265939 A265940 A265941 * A265943 A265944 A265945

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jan 06 2016

STATUS

approved

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Last modified April 20 09:03 EDT 2019. Contains 322307 sequences. (Running on oeis4.)