A337578 O.g.f. A(x) satisfies: [x^n] exp( n*(n+1)^2 * x ) / A(x)^(n+1) = 0 for n > 0.

1, 2, 26, 1756, 301140, 94035272, 45829458720, 31938032357440, 30067763435823664, 36733573872360057568, 56505395303074211175584, 106885908721218962904619840, 243929162811364027276490748288, 661085501644539366377254535077376

Offset

0,2

Comments

It is remarkable that this sequence consists entirely of integers.

Links

Table of n, a(n) for n=0..13.

Formula

Given o.g.f. A(x), define B(x) = A(x/B(x)), then B(x) is the o.g.f. of A337577 and satisfies [x^n] exp( n^2*(n-1) * x/B(x) ) = 0 for n>0.

Example

O.g.f.: A(x) = 1 + 2*x + 26*x^2 + 1756*x^3 + 301140*x^4 + 94035272*x^5 + 45829458720*x^6 + 31938032357440*x^7 + 30067763435823664*x^8 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp( n*(n+1)^2*x ) / A(x)^(n+1) begins:
n=0: [1, -2, -44, -9960, -7049664, -11131647360, -32715852151680, ...];
n=1: [1, 0, -96, -20480, -14247072, -22395261696, -65687348011520, ...];
n=2: [1, 12, 0, -34176, -22928112, -34905615552, -100977330265344, ...];
n=3: [1, 40, 1408, 0, -34275648, -51114811392, -142803802229760, ...];
n=4: [1, 90, 7860, 613000, 0, -71887626240, -199085724252800, ...];
n=5: [1, 168, 27936, 4535040, 663960096, 0, -269327647065600, ...];
n=6: [1, 280, 78064, 21598080, 5858601168, 1443397611264, 0, ...];
n=7: [1, 432, 186240, 80041984, 34200321408, 14371727121408, 5514496883009536, 0, ...]; ...
in which the main diagonal is all zeros after the initial term, illustrating that [x^n] exp( n*(n+1)^2*x ) / A(x)^(n+1) = 0 for n>0.
RELATED SERIES.
Define B(x) = A(x/B(x)), which begins
B(x) = 1 + 2*x + 22*x^2 + 1616*x^3 + 286700*x^4 + 90914400*x^5 + 44673096808*x^6 + 31286975152640*x^7 + ... + A337577(n)*x^n + ...
then the table of coefficients of x^k/k! in exp( n*(n-1)^2 * x/B(x) ) begins:
n=0: [1, 0, 0, 0, 0, 0, 0, ...];
n=1: [1, 0, 0, 0, 0, 0, 0, ...];
n=2: [1, 4, 0, -560, -154880, -137342976, -261610747904, ...];
n=3: [1, 18, 252, 0, -822960, -670328352, -1230620630976, ...];
n=4: [1, 48, 2112, 77760, 0, -2077949952, -3628874151936, ...];
n=5: [1, 100, 9600, 869200, 68473600, 0, -8724419840000, ...];
n=6: [1, 180, 31680, 5423760, 890714880, 130187520000, 0, ...];
n=7: [1, 294, 85260, 24343200, 6817260240, 1850897137824, 453595543361856, 0, ...]; ...
in which the main diagonal is all zeros after the initial term, illustrating that [x^n] exp( n*(n-1)^2 * x/B(x) ) = 0 for n>0.
Also note that B(x) = x/Series_Reversion( x*A(x) ) and A(x) = B(x*A(x)).

Prog

(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); m=#A; A[m] = Vec( exp(m^2*(m-1)*x +x*O(x^m)) / Ser(A)^m )[m]/m); A[n+1]}
for(n=0, 20, print1(a(n), ", "))

Crossrefs

Cf. A337577, A337576, A337458, A337579.

Sequence in context: A363892 A158120 A209916 * A156213 A159318 A318132

Adjacent sequences: A337575 A337576 A337577 * A337579 A337580 A337581

Keyword

nonn

Author

Paul D. Hanna, Sep 02 2020

Status

approved