A181289 -id:A181289 - cp's OEIS frontend

A006637 Expansion of (2 - x)^4/(1 - x)^8.

16, 96, 344, 952, 2241, 4712, 9108, 16488, 28314, 46552, 73788, 113360, 169507, 247536, 354008, 496944, 686052, 932976, 1251568, 1658184, 2172005, 2815384, 3614220, 4598360, 5802030, 7264296, 9029556, 11148064, 13676487, 16678496, 20225392, 24396768, 29281208

Offset: 0

Simon Plouffe

Former name: From generalized Catalan numbers. - G. C. Greubel, Sep 03 2025

H. M. Finucan, Some decompositions of generalized Catalan numbers, pp. 275-293 of Combinatorial Mathematics IX. Proc. Ninth Australian Conference (Brisbane, August 1981). Ed. E. J. Billington, S. Oates-Williams and A. P. Street. Lecture Notes Math., 952. Springer-Verlag, 1982.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Cf. A006633, A006634, A006635, A006636, A181289.

A006637:= func< n | (n+1)*(n+2)*(n+3)*(n+5)*(n+14)*(n^2+31*n+192)/5040 >;
[A006637(n): n in [0..40]]; // G. C. Greubel, Sep 03 2025

Table[(n+1)*(n+2)*(n+3)*(n+5)*(n+14)*(n^2+31*n+192)/7!, {n,0,40}] (* G. C. Greubel, Sep 03 2025 *)

def A006637(n): return (n+1)*(n+2)*(n+3)*(n+5)*(n+14)*(n^2+31*n+192)//5040
print([A006637(n) for n in range(41)]) # G. C. Greubel, Sep 03 2025

G.f.: (2-x)^4/(1-x)^8. - Sean A. Irvine, May 31 2017
a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8). - Wesley Ivan Hurt, Jun 18 2022
From G. C. Greubel, Sep 03 2025: (Start)
a(n) = Sum_{k=0..4} binomial(4, k)*binomial(n+k+3, k+3).
a(n) = (1/7!)*(n+1)*(n+2)*(n+3)*(n+5)*(n+14)*(n^2 + 31*n + 192).
E.g.f.: (1/7!)*(80640 + 403200*x + 423360*x^2 + 161280*x^3 + 27090*x^4 + 2142*x^5 + 77*x^6 + x^7)*exp(x). (End)

a(6) and a(8) corrected and more terms from Sean A. Irvine, May 31 2017

New name by G. C. Greubel, Sep 03 2025

A253722 Triangle read by rows: coefficients of the partition polynomials for the reciprocal of the derivative of a power series, g(x)= 1/h'(x).

Original entry on oeis.org

1, -2, 4, -3, -8, 12, -4, 16, -36, 9, 16, -5, -32, 96, -54, -48, 24, 20, -6, 64, -240, 216, 128, -27, -144, -60, 16, 30, 24, -7, -128, 576, -720, -320, 216, 576, 160, -108, -96, -180, -72, 40, 36, 28, -8

Offset: 0

Tom Copeland, May 02 2015

This entry contains the integer coefficients of the partition polynomials P(n;h_1,h_2,...,h_(n+1)) for the reciprocal g(x) of the derivative of a power series in terms of the coefficients of the power series; i.e., g(x) = 1/[dh(x)/dx] = 1/[h_1 + 2*h_2 * x + 3*h_3 * x^2 + ...] = sum[n>=0, (h_1)^(-(n+1)) * P(n;h_1,...,h_(n+1)) * x^n].

This is a signed refinement of reversed A181289. See A145271, A133437, and A133314 for relations to compositional and multiplicative inversions.

			Let h(x) = h_0 + h_1 * x + h_2 * x^2 + ... . Then g(x) = 1/h'(x) = 1/[h_1 + 2*h_2 * x + 3*h_3 * x^2 + ...] = (h_1)^(-1) P(0;h_1) + (h_1)^(-2) * P(1;h_1,h_2) * x + (h_1)^(-3) * P(2;h_1,h_2,h_3) * x^2 + ... , and, with h_n = (n'), the first few partition polynomials are
P(0;..)=  1
P(1;..)= -2 (2')
P(2;..)=  4 (2')^2 - 3 (3')(1')
P(3;..)= -8 (2')^3 + 12 (3')(2')(1') - 4 (4')(1')^2
P(4;..)= 16 (2')^4 - 36 (2')^2(3')(1') + [9 (3')^2 + 16 (4')(2')](1')^2 - 5 (5')(1')^3
P(5;..)= -32 (2')^5 + 96 (2')^3(3')(1') + [-54 (3')^2(2') - 48 (4')(2')^2](1')^2 + [24 (3')(4') + 20 (5')(2')](1')^3 - 6 (6')(1')^4
P(6;..)= 64 (2')^6 - 240 (2')^4(3')(1') + [216 (3')^2(2') + 128 (4')(2')^3](1')^2 - [27 (3')^3 + 144 (4')(3')(2') + 60 (5')(2')^2](1')^3 + [16 (4')^2 + 30 (5')(3') + 24 (6')(2')](1')^4 - 7 (7')(1')^5

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Cf. A181289, A133437, A145271, A133314, A003480.

rows[n_] := {{1}}~Join~With[{s = 1/(1 + Sum[(k+1) u[k] x^k, {k, n}] + O[x]^(n+1))}, Table[Coefficient[s, x^k Product[u[t], {t, p}]], {k, n}, {p, Reverse@Sort[Sort /@ IntegerPartitions[k]]}]];
rows[7] // Flatten (* Andrey Zabolotskiy, Feb 19 2024 *)

C(v)={my(S=Set(v)); (-1)^(#v)*(#v)!*prod(i=1, #S, my(x=S[i], e=#select(y-> y==x, v)); (x+1)^e/e! )}
row(n)=[C(Vec(p)) | p<-Vecrev(partitions(n))]
{ for(n=0, 7, print(row(n))) } \\ Andrew Howroyd, Feb 19 2024

For the partition (1')^e(1)*(2')^e(2)*...*(n')^e(n) in P(m;...), the unsigned integer coefficient is [e(2)+e(3)+...+e(n)]! * [2^e(2)*3^e(3)*...*n^e(n)]/[e(2)!*e(3)!*...*e(n)!] with the sign determined by (-1)^[e(1) + m].
The partitions of P(m;..) are formed by adding one to each index of the partitions of m of Abramowitz and Stegun's partition table (p. 831; in the reversed order) and appending (1')^e(1) as a factor to obtain a partition of 2m.
Row sums are 1,-2,1,0,0,0,... . Row sums of the unsigned coefficients are A003480.

Row 7 added by Andrey Zabolotskiy, Feb 19 2024

cp's OEIS Frontend

A006637 Expansion of (2 - x)^4/(1 - x)^8.

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