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.

Showing 1-2 of 2 results.

A321087 O.g.f. A(x) satisfies: [x^n] exp(n*A(x)) * (1 - n*x/(1-x)) = 0, for n > 0.

Original entry on oeis.org

1, 2, 7, 37, 256, 2128, 20294, 216213, 2530522, 32165101, 440388103, 6454695553, 100786308221, 1669953080587, 29265149535076, 540884779563305, 10516595791609376, 214625521232021413, 4588068733776013386, 102541337542692407011, 2391813703854249362395, 58130860852912365134992, 1469860403455095402834628, 38611523432412179047238389
Offset: 1

Views

Author

Paul D. Hanna, Oct 27 2018

Keywords

Comments

It is remarkable that this sequence should consist entirely of integers.
Compare to: [x^n] exp(n*G(x)) * (1 - n*x) = 0, for n > 0, when G(x) = x + x*G(x)*G'(x), where G(x)/x is the o.g.f. of A088716.

Examples

			O.g.f.: A(x) = x + 2*x^2 + 7*x^3 + 37*x^4 + 256*x^5 + 2128*x^6 + 20294*x^7 + 216213*x^8 + 2530522*x^9 + 32165101*x^10 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp(-n*A(x)) * (1 - n*x/(1-x)) begins:
n=1: [1, 0, 1, 28, 801, 30256, 1544425, 103604796, 8828789473, ...];
n=2: [1, 0, 0, 32, 1296, 55632, 2987200, 204441120, 17560833024, ...];
n=3: [1, 0, -3, 0, 1161, 67608, 4053645, 290790216, 25525161585, ...];
n=4: [1, 0, -8, -80, 0, 54304, 4333120, 344829888, 31719439360, ...];
n=5: [1, 0, -15, -220, -2655, 0, 3244825, 340694100, 34696521825, ...];
n=6: [1, 0, -24, -432, -7344, -115344, 0, 242169696, 32423666688, ...];
n=7: [1, 0, -35, -728, -14679, -316568, -6439475, 0, 22110305329, ...];
n=8: [1, 0, -48, -1120, -25344, -633792, -17406080, -451234944, 0, ...]; ...
in which the coefficient of x^n in row n forms a diagonal of zeros.
RELATED SERIES.
(a) Differential Equation.
O.g.f. A(x) satisfies: A(x) = x/(1-x) + x*A(x)*A'(x) where
A'(x) = 1 + 4*x + 21*x^2 + 148*x^3 + 1280*x^4 + 12768*x^5 + 142058*x^6 + ...
A(x)*A'(x) = x + 6*x^2 + 36*x^3 + 255*x^4 + 2127*x^5 + 20293*x^6 + 216212*x^7 + 2530521*x^8 + 32165100*x^9 + ...
so that A(x) - x*A(x)*A'(x) = x/(1-x).
(b) Exponentiation.
exp(A(x)) = 1 + x + 5*x^2/2! + 55*x^3/3! + 1129*x^4/4! + 37541*x^5/5! + 1813381*x^6/6! + 118181155*x^7/7! + 9890849585*x^8/8! + ...
exp(-A(x)) = 1 - x - 3*x^2/2! - 31*x^3/3! - 695*x^4/4! - 25221*x^5/5! - 1299779*x^6/6! - 88812907*x^7/7! - 7702826319*x^8/8! + ...

Crossrefs

Cf. A321086, A088716.

Programs

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

Formula

O.g.f. A(x) satisfies: A(x) = x/(1-x) + x*A(x)*A'(x).

A321085 O.g.f. A(x) satisfies: [x^n] exp(-n*A(x)) / (1 - n*x - n*x^2) = 0, for n > 0.

Original entry on oeis.org

1, 2, 6, 36, 330, 4092, 63308, 1165952, 24802704, 596862420, 16003332092, 472588688880, 15231153085304, 531801227022912, 19992702649780800, 805099660439308672, 34574148236596003008, 1577229602738525889984, 76172611622917782581040, 3882814240467405224924960, 208333046141920439476582608, 11737307286279509352537747760
Offset: 1

Views

Author

Paul D. Hanna, Oct 27 2018

Keywords

Comments

It is remarkable that this sequence should consist entirely of integers.
Compare to: [x^n] exp(n*G(x)) * (1 - n*x - n*x^2) = 0, for n > 0, when G(x) = x + x^2 + x*G(x)*G'(x), where G(x)/x is the o.g.f. of A321086.

Examples

			O.g.f.: A(x) = x + 2*x^2 + 6*x^3 + 36*x^4 + 330*x^5 + 4092*x^6 + 63308*x^7 + 1165952*x^8 + 24802704*x^9 + 596862420*x^10 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp(-n*A(x)) / (1 - n*x - n*x^2) begins:
n=1: [1, 0, -1, -28, -819, -39056, -2923925, -317422764, ...];
n=2: [1, 0, 0, -32, -1392, -75552, -5832320, -635767680, ...];
n=3: [1, 0, 3, 0, -1323, -100008, -8542665, -955410984, ...];
n=4: [1, 0, 8, 80, 0, -89024, -10215680, -1248268032, ...];
n=5: [1, 0, 15, 220, 3405, 0, -8752325, -1409888100, ...];
n=6: [1, 0, 24, 432, 9936, 234144, 0, -1176833664, ...];
n=7: [1, 0, 35, 728, 20853, 710248, 23232055, 0, ...];
n=8: [1, 0, 48, 1120, 37632, 1560192, 72348160, 3135469056, 0, ...]; ...
in which the coefficient of x^n in row n forms a diagonal of zeros.
RELATED SERIES.
exp(A(x)) = 1 + x + 5*x^2/2! + 49*x^3/3! + 1081*x^4/4! + 46001*x^5/5! + 3272701*x^6/6! + 345526945*x^7/7! + 50126588849*x^8/8! + ...
exp(-A(x)) = 1 - x - 3*x^2/2! - 25*x^3/3! - 695*x^4/4! - 34401*x^5/5! - 2665019*x^6/6! - 295314937*x^7/7! - 44140455855*x^8/8! + ...

Crossrefs

Cf. A319938, A321086.

Programs

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

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