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.

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A337758 G.f. A(x) satisfies: [x^n] (1 + n*x + n*x^2 - A(x))^(n+1) = 0, for n > 0.

Original entry on oeis.org

1, 3, 8, 41, 284, 2594, 29420, 395845, 6109724, 105714438, 2017696504, 41979555034, 943466064072, 22739452659420, 584304270694436, 15928490898945133, 458761105965272316, 13910124960218668430, 442657291681105692624, 14744175994124292681518, 512800784035081173166088
Offset: 1

Views

Author

Paul D. Hanna, Oct 24 2020

Keywords

Comments

Compare to: [x^n] (1 + n*x - C(x))^(n+1) = 0, for n>0, where C(x) = x + C(x)^2 is a g.f. of the Catalan numbers (A000108).
Compare to: [x^n] (1 + n*x - W(x))^n = 0, for n>0, where W(x) = Sum_{n>=1} (n-1)^(n-1)*x^n/n! = 1 + x/LambertW(-x).

Examples

			G.f.: A(x) = x + 3*x^2 + 8*x^3 + 41*x^4 + 284*x^5 + 2594*x^6 + 29420*x^7 + 395845*x^8 + 6109724*x^9 + 105714438*x^10 + 2017696504*x^11 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in (1 + n*x + n*x^2 - A(x))^(n+1) begins:
n=0: [1, -1, -3, -8, -41, -284, -2594, -29420, -395845, ...];
n=1: [1, 0, -4, -16, -78, -536, -4960, -57048, -775089, ...];
n=2: [1, 3, 0, -29, -171, -1071, -9124, -100590, -1334241, ...];
n=3: [1, 8, 24, 0, -340, -2504, -19032, -189408, -2368430, ...];
n=4: [1, 15, 95, 290, 0, -5327, -46335, -409770, -4606315, ...];
n=5: [1, 24, 252, 1472, 4614, 0, -103528, -1028952, -10296567, ...];
n=6: [1, 35, 546, 4949, 27972, 90244, 0, -2388773, -26537259, ...];
n=7: [1, 48, 1040, 13376, 112344, 627280, 2083504, 0, -63579020, ...];
n=8: [1, 63, 1809, 31260, 360765, 2901258, 16172964, 55276020, 0, ...]; ...
in which the main diagonal is all zeros after the initial term, illustrating that [x^n] (1 + n*x + n*x^2 - A(x))^(n+1) = 0, for n > 0.
ODD TERMS.
The odd terms seem to occur only at positions equal to powers of 2: a(1), a(2), a(4), a(8), a(16), ...; the odd terms begin: [1, 3, 41, 395845, 15928490898945133, 490833755530209698774408313021523960879357, ...].
RELATED SERIES.
B(x) = 1/(1 - A(x)) = 1 + x + 4*x^2 + 15*x^3 + 76*x^4 + 478*x^5 + 3868*x^6 + 39675*x^7 + 498120*x^8 + 7351430*x^9 + 123503516*x^10 + 2309531318*x^11 + ...
where [x^n] (1 + n*(x + x^2)*B(x))^(n+1) / B(x)^(n+1) = 0 for n > 0.
Series_Reversion(A(x)) = x - 3*x^2 + 10*x^3 - 56*x^4 + 268*x^5 - 2104*x^6 + 7636*x^7 - 129976*x^8 - 369988*x^9 - 19147364*x^10 - 279267684*x^11 - ...

Links

Crossrefs

Cf. A338328, A000108, A018900.

Programs

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

Formula

a(n) is odd iff n is a power of 2 (conjecture).
a(n) = 2 (mod 4) iff n is twice the sum of two distinct powers of 2 (conjecture).

A338408 E.g.f. A(x) satisfies: [x^n] (1 + n*x - A(x))^(2*n) = 0, for n > 0.

Original entry on oeis.org

1, 3, 70, 4515, 567576, 116389295, 35111089728, 14574226069095, 7944376570503040, 5494208894263886139, 4694820247236686649600, 4853712224007783889422923, 5968210130160831707746406400, 8605241830169634366425696447655, 14375558607944255605507888571539456
Offset: 1

Views

Author

Paul D. Hanna, Oct 24 2020

Keywords

Comments

Compare to: [x^n] (1 + n*x - W(x))^n = 0, for n>0, where W(x) = Sum_{n>=1} (n-1)^(n-1)*x^n/n! = 1 + x/LambertW(-x).
Compare to: [x^n] (1 + n*x - C(x))^(n+1) = 0, for n>0, where C(x) = x + C(x)^2 is a g.f. of the Catalan numbers (A000108).

Examples

			E.g.f.: A(x) = x + 3*x^2/2! + 70*x^3/3! + 4515*x^4/4! + 567576*x^5/5! + 116389295*x^6/6! + 35111089728*x^7/7! + 14574226069095*x^8/8! + 7944376570503040*x^9/9! + 5494208894263886139*x^10/10! + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in (1 + n*x - A(x))^(2*n) begins:
n=0: [1, 0, 0, 0, 0, 0, 0, 0, ...];
n=1: [1, 0, -6, -140, -8976, -1130952, -232274240, -70128541380, ...];
n=2: [1, 4, 0, -364, -21504, -2530284, -504753152, -149907313980, ...];
n=3: [1, 12, 102, 0, -45960, -5063916, -928551600, -263868802728, ...];
n=4: [1, 24, 480, 7000, 0, -9924168, -1748523008, -457324971720, ...];
n=5: [1, 40, 1410, 42140, 939360, 0, -3259331360, -836926230780, ...];
n=6: [1, 60, 3264, 158220, 6595584, 208807788, 0, -1509806731620, ...];
n=7: [1, 84, 6510, 460936, 29355816, 1626947196, 69489455728, 0, ...]; ...
in which the main diagonal is all zeros after the initial term, illustrating that [x^n] (1 + n*x - A(x))^(2*n) = 0, for n > 0.

Links

Crossrefs

Cf. A338328, A337758, A350366.

Programs

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

Formula

a(n) ~ c * d^n * n!^2 / n^2, where d = (1+r) / ((-1 + exp(r + LambertW(-1, -exp(-r)*r))) * LambertW(-exp(-1-r)*(1+r))) = 8.406107401279769476199925123910168..., r = 0.7545302104650497245839827141610818561001159135034... is the root of the equation r*(1 + r + LambertW(-exp(-1 - r)*(1 + r))) = -(1 + r)*(r + LambertW(-1, -exp(-r)*r)) and c = 0.031468237083... - Vaclav Kotesovec, Aug 12 2021, updated Dec 29 2021
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