A243953 -id:A243953 - cp's OEIS frontend

A127670 Discriminants of Chebyshev S-polynomials A049310.

1, 4, 32, 400, 6912, 153664, 4194304, 136048896, 5120000000, 219503494144, 10567230160896, 564668382613504, 33174037869887488, 2125764000000000000, 147573952589676412928, 11034809241396899282944, 884295678882933431599104, 75613185918270483380568064

Offset: 1

Wolfdieter Lang, Jan 23 2007

a(n-1) is the number of fixed n-cell polycubes that are proper in n-1 dimensions (Barequet et al., 2010).
From Rigoberto Florez, Sep 02 2018: (Start)
a(n-1) is the discriminant of the Morgan-Voyce Fibonacci-type polynomial B(n).
Morgan-Voyce Fibonacci-type polynomials are defined as B(0) = 0, B(1) = 1 and B(n) = (x+2)*B(n-1) - B(n-2) for n > 1.
The absolute value of the discriminant of Fibonacci polynomial F(n) is a(n-1).
Fibonacci polynomials are defined as F(0) = 0, F(1) = 1 and F(n) = x*F(n-1) + F(n-2) for n > 1. (End)
The first 6 values are the dimensions of the polynomial ring in 3n variables xi, yi, zi for 1 <= i <= n modulo the ideal generated by x1^a y1^b z1^c + ... + xn^a yn^b zn^c for 0 < a+b+c <= n (see Fact 2.8.1 in Haiman's paper). - Mike Zabrocki, Dec 31 2019

			n=3: The zeros are [sqrt(2),0,-sqrt(2)]. The Vn(xn[1],...,xn[n]) matrix is [[1,1,1],[sqrt(2),0,-sqrt(2)],[2,0,2]]. The squared determinant is 32 = a(3). - _Wolfdieter Lang_, Aug 07 2011

Gill Barequet, Solomon W. Golomb, and David A. Klarner, Polyominoes. (This is a revision, by G. Barequet, of the chapter of the same title originally written by the late D. A. Klarner for the first edition, and revised by the late S. W. Golomb for the second edition.) Preprint, 2016, http://www.csun.edu/~ctoth/Handbook/chap14.pdf.
G. Barequet and M. Shalah, Automatic Proofs for Formulae Enumerating Proper Polycubes, 31st International Symposium on Computational Geometry (SoCG'15). Editors: Lars Arge and János Pach; pp. 19-22, 2015.
Theodore J. Rivlin, Chebyshev polynomials: from approximation theory to algebra and number theory, 2. ed., Wiley, New York, 1990; p. 219 for T and U polynomials.

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Andrei Asinowski, Gill Barequet, Ronnie Barequet, and Gunter Rote, Proper n-Cell Polycubes in n - 3 Dimensions, Journal of Integer Sequences, Vol. 15 (2012), #12.8.4.
Mohammad K. Azarian, On the Hyperfactorial Function, Hypertriangular Function, and the Discriminants of Certain Polynomials, International Journal of Pure and Applied Mathematics, Vol. 36, No. 2, 2007, pp. 251-257. Mathematical Reviews, MR2312537. Zentralblatt MATH, Zbl 1133.11012. See Th. 1. [From _N. J. A. Sloane_, Oct 16 2010]
R. Barequet, G. Barequet, and G. Rote, Formulae and growth rates of high-dimensional polycubes, Combinatorica 30 (2010), pp. 257-275.
Rigoberto Flórez, Robinson Higuita, and Antara Mukherjee, Star of David and other patterns in the Hosoya-like polynomials triangles, Journal of Integer Sequences, Vol. 21 (2018), Article 18.4.6.
Rigoberto Flórez, Robinson Higuita, and Antara Mukherjee, Characterization of the strong divisibility property for generalized Fibonacci polynomials, Integers, 18 (2018), Paper No. A14.
Rigoberto Flórez, Robinson Higuita, and Alexander Ramírez, The resultant, the discriminant, and the derivative of generalized Fibonacci polynomials, arXiv:1808.01264 [math.NT], 2018.
Rigoberto Flórez, N. McAnally, and A. Mukherjees, Identities for the generalized Fibonacci polynomial, Integers, 18B (2018), Paper No. A2.
M. Haiman, Conjectures on the quotient ring by diagonal invariants, preprint, 1993.
M. Haiman, Conjectures on the quotient ring by diagonal invariants, J. Algebraic Combin. 3 (1994), no. 1, 17--76.
O. Khorunzhiy, Enumeration of tree-type diagrams assembled from oriented chains of edges, arXiv:2207.00766 [math.CO], 2022.
Andrew Snowden, Measures for the colored circle, arXiv:2302.08699 [math.CO], 2023.
Eric Weisstein's World of Mathematics, Discriminant
Eric Weisstein's World of Mathematics, Morgan-Voyce Polynomials
Eric Weisstein's World of Mathematics, Fibonacci Polynomial

Cf. A007701 (T-polynomials), A086804 (U-polynomials), A171860 and A191092 (fixed n-cell polycubes proper in n-2 and n-3 dimensions, resp.).
Cf. A243953, A006645, A001629, A001871, A006645, A007701, A045618, A045925, A093967, A193678, A317404, A317405, A317408, A317451, A318184, A318197.
A317403 is essentially the same sequence.
Diagonal 1 of A195739.

[((n+1)^n/(n+1)^2)*2^n: n in [1..20]]; // Vincenzo Librandi, Jun 23 2014

Table[((n + 1)^n)/(n + 1)^2 2^n, {n, 1, 30}] (* Vincenzo Librandi, Jun 23 2014 *)

a(n) = ((n+1)^(n-2))*2^n, n >= 1.
a(n) = (Det(Vn(xn[1],...,xn[n])))^2 with the determinant of the Vandermonde matrix Vn with elements (Vn)i,j:= xn[i]^j, i=1..n, j=0..n-1 and xn[i]:=2*cos(Pi*i/(n+1)), i=1..n, are the zeros of S(n,x):=U(n,x/2).
a(n) = ((-1)^(n*(n-1)/2))*Product_{j=1..n} ((d/dx)S(n,x)|_{x=xn[j]}), n >= 1, with the zeros xn[j], j=1..n, given above.
a(n) = A007830(n-2)*A000079(n), n >= 2. - Omar E. Pol, Aug 27 2011
E.g.f.: -LambertW(-2*x)*(2+LambertW(-2*x))/(4*x). - Vaclav Kotesovec, Jun 22 2014

Slightly edited by Gill Barequet, May 24 2011

A251573 E.g.f.: exp(3xG(x)^2) / G(x)^2 where G(x) = 1 + x*G(x)^3 is the g.f. of A001764.

Original entry on oeis.org

1, 1, 3, 21, 261, 4833, 120303, 3778029, 143531433, 6404711553, 328447585179, 19037277446949, 1230842669484717, 87829738967634849, 6856701559496841159, 581343578623728854397, 53196439113856500195537, 5225543459274294130169601, 548468830470032135590262067, 61258398893626609968686844597

Offset: 0

Paul D. Hanna, Dec 06 2014

nonn

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 21*x^3/3! + 261*x^4/4! + 4833*x^5/5! +...
such that A(x) = exp(3*x*G(x)^2) / G(x)^2
where G(x) = 1 + x*G(x)^3 is the g.f. of A001764:
G(x) = 1 + x + 3*x^2 + 12*x^3 + 55*x^4 + 273*x^5 + 1428*x^6 +...
The e.g.f. satisfies:
A(x) = 1 + x/A(x) + 5*x^2/(2!*A(x)^2) + 54*x^3/(3!*A(x)^3) + 945*x^4/(4!*A(x)^4) + 23328*x^5/(5!*A(x)^5) + 750141*x^6/(6!*A(x)^6) + 29859840*x^7/(7!*A(x)^7) +...+ 3^(n-1)*(n+1)^(n-3)*(n+3) * x^n/(n!*A(x)^n) +...
Note that
A'(x) = exp(3*x*G(x)^2) = 1 + 3*x + 21*x^2/2! + 261*x^3/3! + 4833*x^4/4! +...
LOGARITHMIC DERIVATIVE.
The logarithm of the e.g.f. begins:
log(A(x)) = x + 2*x^2/2 + 7*x^3/3 + 30*x^4/4 + 143*x^5/5 +...
and so A'(x)/A(x) = G(x)^2.
TABLE OF POWERS OF E.G.F.
Form a table of coefficients of x^k/k! in A(x)^n as follows.
n=1: [1, 1,  3,   21,   261,   4833,  120303,   3778029, ...];
n=2: [1, 2,  8,   60,   744,  13536,  330912,  10232928, ...];
n=3: [1, 3, 15,  123,  1557,  28179,  680427,  20771235, ...];
n=4: [1, 4, 24,  216,  2832,  51552, 1237248,  37404288, ...];
n=5: [1, 5, 35,  345,  4725,  87285, 2094975,  62949825, ...];
n=6: [1, 6, 48,  516,  7416, 139968, 3378528, 101278944, ...];
n=7: [1, 7, 63,  735, 11109, 215271, 5250987, 157613463, ...];
n=8: [1, 8, 80, 1008, 16032, 320064, 7921152, 238878720, ...]; ...
in which the main diagonal begins (see A251583):
[1, 2, 15, 216, 4725, 139968, 5250987, 238878720, ...]
and is given by the formula:
[x^n/n!] A(x)^(n+1) = 3^(n-1) * (n+1)^(n-2) * (n+3) for n>=0.

Paul D. Hanna, Table of n, a(n) for n = 0..200

Cf. A251583, A251663, A001764, A006013.

Cf. Variants: A243953, A251574, A251575, A251576, A251577, A251578, A251579, A251580.

Flatten[{1,1,Table[Sum[3^k * n!/k! * Binomial[3*n-k-3, n-k] * (k-1)/(n-1),{k,0,n}],{n,2,20}]}] (* Vaclav Kotesovec, Dec 07 2014 *)
Flatten[{1,1,RecurrenceTable[{27*(n-2)*a[n-2]-3*(3*n-8)*(15-13*n+3*n^2)*a[n-1]+2*(n-3)*(2*n-3)*a[n]==0,a[2]==3,a[3]==21},a,{n,20}]}] (* Vaclav Kotesovec, Dec 07 2014 *)

{a(n) = local(G=1);for(i=1,n,G=1+x*G^3 +x*O(x^n)); n!*polcoeff(exp(3*x*G^2)/G^2, n)}
for(n=0, 20, print1(a(n), ", "))

{a(n) = if(n==0||n==1, 1, sum(k=0, n, 3^k * n!/k! * binomial(3*n-k-3,n-k) * (k-1)/(n-1) ))}
for(n=0, 20, print1(a(n), ", "))

Let G(x) = 1 + x*G(x)^3 be the g.f. of A001764, then the e.g.f. A(x) of this sequence satisfies:
(1) A'(x)/A(x) = G(x)^2.
(2) A'(x) = exp(3*x*G(x)^2).
(3) A(x) = exp( Integral G(x)^2 dx ).
(4) A(x) = exp( Sum_{n>=1} A006013(n-1)*x^n/n ), where A006013(n-1) = binomial(3*n-2,n)/(2*n-1).
(5) A(x) = F(x/A(x)) where F(x) is the e.g.f. of A251583.
(6) A(x) = Sum_{n>=0} A251583(n)*(x/A(x))^n/n! where A251583(n) = 3^(n-1) * (n+1)^(n-3) * (n+3).
(7) [x^n/n!] A(x)^(n+1) = 3^(n-1) * (n+1)^(n-2) * (n+3).
a(n) = Sum_{k=0..n} 3^k * n!/k! * binomial(3*n-k-3, n-k) * (k-1)/(n-1) for n>1.
Recurrence (for n>3): 2*(n-3)*(2*n-3)*a(n) = 3*(3*n-8)*(3*n^2 - 13*n + 15)*a(n-1) - 27*(n-2)*a(n-2). - Vaclav Kotesovec, Dec 07 2014
a(n) ~ 3^(3*n-7/2) * n^(n-2) / (2^(2*n-5/2) * exp(n-1)). - Vaclav Kotesovec, Dec 07 2014

A251574 E.g.f.: exp(4xG(x)^3) / G(x)^3 where G(x) = 1 + x*G(x)^4 is the g.f. of A002293.

Original entry on oeis.org

1, 1, 4, 40, 712, 18784, 663424, 29480896, 1581976960, 99585422848, 7198258087936, 587699970912256, 53497834761985024, 5372784803063664640, 590164397145095421952, 70386834555048578596864, 9058611906733586004803584, 1251310862246447324484468736, 184665445630564847038730076160

Offset: 0

Paul D. Hanna, Dec 06 2014

nonn

			E.g.f.: A(x) = 1 + x + 4*x^2/2! + 40*x^3/3! + 712*x^4/4! + 18784*x^5/5! +...
such that A(x) = exp(4*x*G(x)^3) / G(x)^3
where G(x) = 1 + x*G(x)^4 is the g.f. of A002293:
G(x) = 1 + x + 4*x^2 + 22*x^3 + 140*x^4 + 969*x^5 + 7084*x^6 +...
Note that
A'(x) = exp(4*x*G(x)^3) = 1 + 4*x + 40*x^2/2! + 712*x^3/3! + 18784*x^4/4! +...
LOGARITHMIC DERIVATIVE.
The logarithm of the e.g.f. begins:
log(A(x)) = x + 3*x^2/2 + 15*x^3/3 + 91*x^4/4 + 612*x^5/5 +...
and so A'(x)/A(x) = G(x)^3.
TABLE OF POWERS OF E.G.F.
Form a table of coefficients of x^k/k! in A(x)^n as follows.
n=1: [1, 1,  4,   40,   712,  18784,   663424,   29480896, ...];
n=2: [1, 2, 10,  104,  1840,  47888,  1669696,   73399040, ...];
n=3: [1, 3, 18,  198,  3528,  91152,  3146256,  136990656, ...];
n=4: [1, 4, 28,  328,  5944, 153376,  5257024,  227057728, ...];
n=5: [1, 5, 40,  500,  9280, 240440,  8209600,  352337600, ...];
n=6: [1, 6, 54,  720, 13752, 359424, 12263184,  523933056, ...];
n=7: [1, 7, 70,  994, 19600, 518728, 17737216,  755807920, ...];
n=8: [1, 8, 88, 1328, 27088, 728192, 25020736, 1065353216, ...]; ...
in which the main diagonal begins (see A251584):
[1, 2, 18, 328, 9280, 359424, 17737216, 1065353216, ...]
and is given by the formula:
[x^n/n!] A(x)^(n+1) = 4^(n-2) * (n+1)^(n-3) * (3*n^2 + 13*n + 16) for n>=0.

Cf. A251584, A251664, A002293, A006632.

Cf. Variants: A243953, A251573, A251575, A251576, A251577, A251578, A251579, A251580.

Flatten[{1,1,Table[Sum[4^k * n!/k! * Binomial[4*n-k-4, n-k] * (k-1)/(n-1),{k,0,n}],{n,2,20}]}] (* Vaclav Kotesovec, Dec 07 2014 *)

{a(n) = local(G=1);for(i=1,n,G=1+x*G^4 +x*O(x^n)); n!*polcoeff(exp(4*x*G^3)/G^3, n)}
for(n=0, 20, print1(a(n), ", "))

{a(n) = if(n==0||n==1, 1, sum(k=0, n, 4^k * n!/k! * binomial(4*n-k-4,n-k) * (k-1)/(n-1) ))}
for(n=0, 20, print1(a(n), ", "))

Let G(x) = 1 + x*G(x)^4 be the g.f. of A002293, then the e.g.f. A(x) of this sequence satisfies:
(1) A'(x)/A(x) = G(x)^3.
(2) A'(x) = exp(4*x*G(x)^3).
(3) A(x) = exp( Integral G(x)^3 dx ).
(4) A(x) = exp( Sum_{n>=1} A006632(n)*x^n/n ), where A006632(n) = binomial(4*n-2,n)/(3*n-1).
(5) A(x) = F(x/A(x)) where F(x) is the e.g.f. of A251584.
(6) A(x) = Sum_{n>=0} A251584(n)*(x/A(x))^n/n! and
(7) [x^n/n!] A(x)^(n+1) = (n+1)*A251584(n),
where A251584(n) = 4^(n-2) * (n+1)^(n-4) * (3*n^2 + 13*n + 16).
a(n) = Sum_{k=0..n} 4^k * n!/k! * binomial(4*n-k-4, n-k) * (k-1)/(n-1) for n>1.
Recurrence: 3*(3*n-5)*(3*n-4)*(4*n^2 - 23*n + 34)*a(n) = 8*(128*n^5 - 1440*n^4 + 6520*n^3 - 14906*n^2 + 17289*n - 8190)*a(n-1) + 256*(4*n^2 - 15*n + 15)*a(n-2). - Vaclav Kotesovec, Dec 07 2014
a(n) ~ 2^(8*n-9) * n^(n-2) / (3^(3*n-7/2) * exp(n-1)). - Vaclav Kotesovec, Dec 07 2014

A251575 E.g.f.: exp(5xG(x)^4) / G(x)^4 where G(x) = 1 + x*G(x)^5 is the g.f. of A002294.

Original entry on oeis.org

1, 1, 5, 65, 1505, 51505, 2354725, 135258625, 9373203425, 761486105825, 71001537157925, 7475144493546625, 877222642396170625, 113551974107296500625, 16073867927431440597125, 2470217878902686107522625, 409596824402404827033730625, 72890993386914239524503090625, 13857243751694786173837746653125

Offset: 0

Paul D. Hanna, Dec 06 2014

nonn

			E.g.f.: A(x) = 1 + x + 5*x^2/2! + 65*x^3/3! + 1505*x^4/4! + 51505*x^5/5! +...
such that A(x) = exp(5*x*G(x)^4) / G(x)^4
where G(x) = 1 + x*G(x)^5 is the g.f. of A002294:
G(x) = 1 + x + 5*x^2 + 35*x^3 + 285*x^4 + 2530*x^5 + 23751*x^6 +...
Note that
A'(x) = exp(5*x*G(x)^4) = 1 + 5*x + 65*x^2/2! + 1505*x^3/3! + 51505*x^4/4! +...
LOGARITHMIC DERIVATIVE.
The logarithm of the e.g.f. begins:
log(A(x)) = x + 4*x^2/2 + 26*x^3/3 + 204*x^4/4 + 1771*x^5/5 +...
and so A'(x)/A(x) = G(x)^4.
TABLE OF POWERS OF E.G.F.
Form a table of coefficients of x^k/k! in A(x)^n as follows.
n=1: [1, 1,  5,   65,  1505,   51505,  2354725,  135258625, ...];
n=2: [1, 2, 12,  160,  3680,  124560,  5637760,  321147200, ...];
n=3: [1, 3, 21,  291,  6705,  225315, 10112805,  571694355, ...];
n=4: [1, 4, 32,  464, 10784,  361120, 16101760,  904145920, ...];
n=5: [1, 5, 45,  685, 16145,  540645, 23993725, 1339552925, ...];
n=6: [1, 6, 60,  960, 23040,  774000, 34254720, 1903435200, ...];
n=7: [1, 7, 77, 1295, 31745, 1072855, 47438125, 2626525615, ...];
n=8: [1, 8, 96, 1696, 42560, 1450560, 64195840, 3545600000, ...]; ...
in which the main diagonal begins (see A251585):
[1, 2, 21, 464, 16145, 774000, 47438125, 3545600000, ...]
and is given by the formula:
[x^n/n!] A(x)^(n+1) = 5^(n-3) * (n+1)^(n-4) * (16*n^3 + 87*n^2 + 172*n + 125) for n>=0.

Cf. A251585, A251665, A002294, A118971.

Cf. Variants: A243953, A251573, A251574, A251576, A251577, A251578, A251579, A251580.

Flatten[{1,1,Table[Sum[5^k * n!/k! * Binomial[5*n-k-5, n-k] * (k-1)/(n-1),{k,0,n}],{n,2,20}]}] (* Vaclav Kotesovec, Dec 07 2014 *)

{a(n) = local(G=1);for(i=1,n,G=1+x*G^5 +x*O(x^n)); n!*polcoeff(exp(5*x*G^4)/G^4, n)}
for(n=0, 20, print1(a(n), ", "))

{a(n) = if(n==0||n==1, 1, sum(k=0, n, 5^k * n!/k! * binomial(5*n-k-5,n-k) * (k-1)/(n-1) ))}
for(n=0, 20, print1(a(n), ", "))

Let G(x) = 1 + x*G(x)^5 be the g.f. of A002294, then the e.g.f. A(x) of this sequence satisfies:
(1) A'(x)/A(x) = G(x)^4.
(2) A'(x) = exp(5*x*G(x)^4).
(3) A(x) = exp( Integral G(x)^4 dx ).
(4) A(x) = exp( Sum_{n>=1} A118971(n-1)*x^n/n ), where A118971(n-1) = binomial(5*n-2,n)/(4*n-1).
(5) A(x) = F(x/A(x)) where F(x) is the e.g.f. of A251585.
(6) A(x) = Sum_{n>=0} A251585(n)*(x/A(x))^n/n! and
(7) [x^n/n!] A(x)^(n+1) = (n+1)*A251585(n)
where A251585(n) = 5^(n-3) * (n+1)^(n-5) * (16*n^3 + 87*n^2 + 172*n + 125).
a(n) = Sum_{k=0..n} 5^k * n!/k! * binomial(5*n-k-5, n-k) * (k-1)/(n-1) for n>1.
Recurrence: 8*(2*n-3)*(4*n-7)*(4*n-5)*(25*n^3 - 210*n^2 + 598*n - 581)*a(n) = 5*(15625*n^7 - 240625*n^6 + 1592500*n^5 - 5883125*n^4 + 13135350*n^3 - 17781015*n^2 + 13566657*n - 4523904)*a(n-1) - 3125*(25*n^3 - 135*n^2 + 253*n - 168)*a(n-2). - Vaclav Kotesovec, Dec 07 2014
a(n) ~ 5^(5*n-11/2) * n^(n-2) / (2^(8*n-9) * exp(n-1)). - Vaclav Kotesovec, Dec 07 2014

A251576 E.g.f.: exp(6xG(x)^5) / G(x)^5 where G(x) = 1 + x*G(x)^6 is the g.f. of A002295.

Original entry on oeis.org

1, 1, 6, 96, 2736, 115056, 6455376, 454666176, 38610711936, 3842344221696, 438721154343936, 56549927146392576, 8123473514799876096, 1287034084022760677376, 222964032114987212998656, 41930788886197036399190016, 8507629742037888427727486976, 1852490637585980898960109142016

Offset: 0

Paul D. Hanna, Dec 06 2014

nonn

			E.g.f.: A(x) = 1 + x + 6*x^2/2! + 96*x^3/3! + 2736*x^4/4! + 115056*x^5/5! +...
such that A(x) = exp(6*x*G(x)^5) / G(x)^5
where G(x) = 1 + x*G(x)^6 is the g.f. of A002295:
G(x) = 1 + x + 6*x^2 + 51*x^3 + 506*x^4 + 5481*x^5 + 62832*x^6 +...
Note that
A'(x) = exp(6*x*G(x)^5) = 1 + 6*x + 96*x^2/2! + 2736*x^3/3! +...
LOGARITHMIC DERIVATIVE.
The logarithm of the e.g.f. begins:
log(A(x)) = x + 5*x^2/2 + 40*x^3/3 + 385*x^4/4 + 4095*x^5/5 + 46376*x^6/6 +...
and so A'(x)/A(x) = G(x)^5.
TABLE OF POWERS OF E.G.F.
Form a table of coefficients of x^k/k! in A(x)^n as follows.
n=1: [1, 1,   6,   96,  2736,  115056,   6455376,  454666176, ...];
n=2: [1, 2,  14,  228,  6456,  268992,  14968224, 1047087648, ...];
n=3: [1, 3,  24,  402, 11376,  470808,  26011584, 1808151552, ...];
n=4: [1, 4,  36,  624, 17736,  730944,  40143456, 2774490624, ...];
n=5: [1, 5,  50,  900, 25800, 1061400,  58017600, 3989340000, ...];
n=6: [1, 6,  66, 1236, 35856, 1475856,  80395056, 5503484736, ...];
n=7: [1, 7,  84, 1638, 48216, 1989792, 108156384, 7376303088, ...];
n=8: [1, 8, 104, 2112, 63216, 2620608, 142314624, 9676910592, ...]; ...
in which the main diagonal begins (see A251586):
[1, 2, 24, 624, 25800, 1475856, 108156384, 9676910592, ...]
and is given by the formula:
[x^n/n!] A(x)^(n+1) = 6^(n-4) * (n+1)^(n-5) * (125*n^4 + 810*n^3 + 2095*n^2 + 2586*n + 1296) for n>=0.

Cf. A251586, A251666, A130564, A002295.

Cf. Variants: A243953, A251573, A251574, A251575, A251577, A251578, A251579, A251580.

Flatten[{1,1,Table[Sum[6^k * n!/k! * Binomial[6*n-k-6, n-k] * (k-1)/(n-1),{k,0,n}],{n,2,20}]}] (* Vaclav Kotesovec, Dec 07 2014 *)

{a(n) = local(G=1);for(i=1,n,G=1+x*G^6 +x*O(x^n)); n!*polcoeff(exp(6*x*G^5)/G^5, n)}
for(n=0, 20, print1(a(n), ", "))

{a(n) = if(n==0||n==1, 1, sum(k=0, n, 6^k * n!/k! * binomial(6*n-k-6,n-k) * (k-1)/(n-1) ))}
for(n=0, 20, print1(a(n), ", "))

Let G(x) = 1 + x*G(x)^6 be the g.f. of A002295, then the e.g.f. A(x) of this sequence satisfies:
(1) A'(x)/A(x) = G(x)^5.
(2) A'(x) = exp(6*x*G(x)^5).
(3) A(x) = exp( Integral G(x)^5 dx ).
(4) A(x) = exp( Sum_{n>=1} A130564(n)*x^n/n ), where A130564(n) = binomial(6*n-2,n)/(5*n-1).
(5) A(x) = F(x/A(x)) where F(x) is the e.g.f. of A251586.
(6) A(x) = Sum_{n>=0} A251586(n)*(x/A(x))^n/n! and
(7) [x^n/n!] A(x)^(n+1) = (n+1)*A251586(n),
where A251586(n) = 6^(n-4) * (n+1)^(n-6) * (125*n^4 + 810*n^3 + 2095*n^2 + 2586*n + 1296).
a(n) = Sum_{k=0..n} 6^k * n!/k! * binomial(6*n-k-6, n-k) * (k-1)/(n-1) for n>1.
Recurrence: 5*(5*n-9)*(5*n-8)*(5*n-7)*(5*n-6)*(9*n^4 - 99*n^3 + 413*n^2 - 777*n + 559)*a(n) = 72*(5832*n^9 - 113724*n^8 + 986580*n^7 - 5003586*n^6 + 16373448*n^5 - 35916483*n^4 + 52931854*n^3 - 50678109*n^2 + 28701206*n - 7357350)*a(n-1) + 46656*(9*n^4 - 63*n^3 + 170*n^2 - 212*n + 105)*a(n-2). - Vaclav Kotesovec, Dec 07 2014
a(n) ~ 6^(6*(n-1)-1/2) / 5^(5*(n-1)-1/2) * n^(n-2) / exp(n-1). - Vaclav Kotesovec, Dec 07 2014

A251577 E.g.f.: exp(7xG(x)^6) / G(x)^6 where G(x) = 1 + x*G(x)^7 is the g.f. of A002296.

Original entry on oeis.org

1, 1, 7, 133, 4501, 224497, 14926387, 1245099709, 125177105641, 14743403405857, 1991987858095039, 303781606238806549, 51624122993243471293, 9674836841745014156497, 1982441139367342976694379, 440946185623028320815311053, 105810290178441439797537070033, 27247415403508413760437930799681

Offset: 0

Paul D. Hanna, Dec 06 2014

nonn

			E.g.f.: A(x) = 1 + x + 7*x^2/2! + 133*x^3/3! + 4501*x^4/4! + 224497*x^5/5! +...
such that A(x) = exp(7*x*G(x)^6) / G(x)^6
where G(x) = 1 + x*G(x)^7 is the g.f. of A002296:
G(x) = 1 + x + 7*x^2 + 70*x^3 + 819*x^4 + 10472*x^5 + 141778*x^6 +...
Note that
A'(x) = exp(7*x*G(x)^6) = 1 + 7*x + 133*x^2/2! + 4501*x^3/3! +...
LOGARITHMIC DERIVATIVE.
The logarithm of the e.g.f. begins:
log(A(x)) = x + 6*x^2/2 + 57*x^3/3 + 650*x^4/4 + 8184*x^5/5 + 109668*x^6/6 +...
and so A'(x)/A(x) = G(x)^6.
TABLE OF POWERS OF E.G.F.
Form a table of coefficients of x^k/k! in A(x)^n as follows.
n=1: [1, 1,  7,   133,  4501,  224497,  14926387,  1245099709, ...];
n=2: [1, 2,  16,  308, 10360,  512624,  33845728,  2807075264, ...];
n=3: [1, 3,  27,  531, 17829,  876771,  57529143,  4745597787, ...];
n=4: [1, 4,  40,  808, 27184, 1331008,  86864512,  7129675840, ...];
n=5: [1, 5,  55, 1145, 38725, 1891205, 122869075, 10038831425, ...];
n=6: [1, 6,  72, 1548, 52776, 2575152, 166702752, 13564381824, ...];
n=7: [1, 7,  91, 2023, 69685, 3402679, 219682183, 17810832319, ...];
n=8: [1, 8, 112, 2576, 89824, 4395776, 283295488, 22897384832, ...]; ...
in which the main diagonal begins (see A251587):
[1, 2, 27, 808, 38725, 2575152, 219682183, 22897384832, ...]
and is given by the formula:
[x^n/n!] A(x)^(n+1) = 7^(n-5) * (n+1)^(n-6) * (1296*n^5 + 9720*n^4 + 30555*n^3 + 50665*n^2 + 44621*n + 16807) for n>=0.

Cf. A251587, A251667, A002296, A130565.

Cf. Variants: A243953, A251573, A251574, A251575, A251576, A251578, A251579, A251580.

Flatten[{1,1,Table[Sum[7^k * n!/k! * Binomial[7*n-k-7, n-k] * (k-1)/(n-1),{k,0,n}],{n,2,20}]}] (* Vaclav Kotesovec, Dec 07 2014 *)

{a(n) = local(G=1);for(i=1,n,G=1+x*G^7 +x*O(x^n)); n!*polcoeff(exp(7*x*G^6)/G^6, n)}
for(n=0, 20, print1(a(n), ", "))

{a(n) = if(n==0|n==1, 1, sum(k=0, n, 7^k * n!/k! * binomial(7*n-k-7,n-k) * (k-1)/(n-1) ))}
for(n=0, 20, print1(a(n), ", "))

Let G(x) = 1 + x*G(x)^7 be the g.f. of A002296, then the e.g.f. A(x) of this sequence satisfies:
(1) A'(x)/A(x) = G(x)^6.
(2) A'(x) = exp(7*x*G(x)^6).
(3) A(x) = exp( Integral G(x)^6 dx ).
(4) A(x) = exp( Sum_{n>=1} A130565(n)*x^n/n ), where A130565(n) = binomial(7*n-2,n)/(6*n-1).
(5) A(x) = F(x/A(x)) where F(x) is the e.g.f. of A251587.
(6) A(x) = Sum_{n>=0} A251587(n)*(x/A(x))^n/n! and
(7) [x^n/n!] A(x)^(n+1) = (n+1)*A251587(n),
where A251587(n) = 7^(n-5) * (n+1)^(n-7) * (1296*n^5 + 9720*n^4 + 30555*n^3 + 50665*n^2 + 44621*n + 16807).
a(n) = Sum_{k=0..n} 7^k * n!/k! * binomial(7*n-k-7, n-k) * (k-1)/(n-1) for n>1.
Recurrence: 72*(2*n-3)*(3*n-5)*(3*n-4)*(6*n-11)*(6*n-7)*(2401*n^5 - 32585*n^4 + 178311*n^3 - 492779*n^2 + 689623*n - 392491)*a(n) = 7*(282475249*n^11 - 6658345155*n^10 + 71339412375*n^9 - 458968749330*n^8 + 1971937124661*n^7 - 5947597074909*n^6 + 12867618998885*n^5 - 20002508046570*n^4 + 21938241804255*n^3 - 16207858252075*n^2 + 7281095411817*n - 1512276480000)*a(n-1) - 823543*(2401*n^5 - 20580*n^4 + 71981*n^3 - 129346*n^2 + 120663*n - 47520)*a(n-2). - Vaclav Kotesovec, Dec 07 2014
a(n) ~ 7^(7*(n-1)-1/2) / 6^(6*(n-1)-1/2) * n^(n-2) / exp(n-1). - Vaclav Kotesovec, Dec 07 2014

A251578 E.g.f.: exp(8xG(x)^7) / G(x)^7 where G(x) = 1 + x*G(x)^8 is the g.f. of A007556.

Original entry on oeis.org

1, 1, 8, 176, 6896, 397888, 30584128, 2948178304, 342418882688, 46582810477568, 7268517454045184, 1279982790328858624, 251155319283837571072, 54344039464582833577984, 12855960226911391575670784, 3301167001281829056285458432, 914476489427649778704952819712

Offset: 0

Paul D. Hanna, Dec 06 2014

nonn

			E.g.f.: A(x) = 1 + x + 8*x^2/2! + 176*x^3/3! + 6896*x^4/4! + 397888*x^5/5! +...
such that A(x) = exp(8*x*G(x)^7) / G(x)^7
where G(x) = 1 + x*G(x)^8 is the g.f. of A007556:
G(x) = 1 + x + 8*x^2 + 92*x^3 + 1240*x^4 + 18278*x^5 + 285384*x^6 +...
Note that
A'(x) = exp(8*x*G(x)^7) = 1 + 8*x + 176*x^2/2! + 6896*x^3/3! +...
LOGARITHMIC DERIVATIVE.
The logarithm of the e.g.f. begins:
log(A(x)) = x + 7*x^2/2 + 77*x^3/3 + 1015*x^4/4 + 14763*x^5/5 +...
and so A'(x)/A(x) = G(x)^7.
TABLE OF POWERS OF E.G.F.
Form a table of coefficients of x^k/k! in A(x)^n as follows.
n=1: [1, 1,   8,  176,   6896,  397888,  30584128,  2948178304, ...];
n=2: [1, 2,  18,  400,  15584,  892896,  68217472,  6543183488, ...];
n=3: [1, 3,  30,  678,  26352, 1501344, 114073632, 10890011520, ...];
n=4: [1, 4,  44, 1016,  39512, 2241472, 169479808, 16107837568, ...];
n=5: [1, 5,  60, 1420,  55400, 3133560, 235931200, 22331561600, ...];
n=6: [1, 6,  78, 1896,  74376, 4200048, 315106128, 29713474944, ...];
n=7: [1, 7,  98, 2450,  96824, 5465656, 408881872, 38425052848, ...];
n=8: [1, 8, 120, 3088, 123152, 6957504, 519351232, 48658878080, ...]; ...
in which the main diagonal begins (see A251587):
[1, 2, 30, 1016, 55400, 4200048, 408881872, 48658878080, ...]
and is given by the formula:
[x^n/n!] A(x)^(n+1) = 8^(n-6) * (n+1)^(n-7) * (16807*n^6 + 143031*n^5 + 525875*n^4 + 1074745*n^3 + 1294846*n^2 + 876856*n + 262144) for n>=0.

Cf. A251588, A251668, A007556, A234466.

Cf. Variants: A243953, A251573, A251574, A251575, A251576, A251577, A251579, A251580.

Flatten[{1,1,Table[Sum[8^k * n!/k! * Binomial[8*n-k-8, n-k] * (k-1)/(n-1),{k,0,n}],{n,2,20}]}] (* Vaclav Kotesovec, Dec 07 2014 *)

{a(n) = local(G=1);for(i=1,n,G=1+x*G^8 +x*O(x^n)); n!*polcoeff(exp(8*x*G^7)/G^7, n)}
for(n=0, 20, print1(a(n), ", "))

{a(n) = if(n==0|n==1, 1, sum(k=0, n, 8^k * n!/k! * binomial(8*n-k-8,n-k) * (k-1)/(n-1) ))}
for(n=0, 20, print1(a(n), ", "))

Let G(x) = 1 + x*G(x)^8 be the g.f. of A007556, then the e.g.f. A(x) of this sequence satisfies:
(1) A'(x)/A(x) = G(x)^7.
(2) A'(x) = exp(8*x*G(x)^7).
(3) A(x) = exp( Integral G(x)^7 dx ).
(4) A(x) = exp( Sum_{n>=1} A234466(n-1)*x^n/n ), where A234466(n-1)(n) = binomial(8*n-2,n)/(7*n-1).
(5) A(x) = F(x/A(x)) where F(x) is the e.g.f. of A251588.
(6) A(x) = Sum_{n>=0} A251588(n)*(x/A(x))^n/n! and
(7) [x^n/n!] A(x)^(n+1) = (n+1)*A251588(n),
where A251588(n) = 8^(n-6) * (n+1)^(n-8) * (16807*n^6 + 143031*n^5 + 525875*n^4 + 1074745*n^3 + 1294846*n^2 + 876856*n + 262144).
a(n) = Sum_{k=0..n} 8^k * n!/k! * binomial(8*n-k-8, n-k) * (k-1)/(n-1) for n>1.
Recurrence: 7*(7*n-13)*(7*n-12)*(7*n-11)*(7*n-10)*(7*n-9)*(7*n-8)*(4096*n^6 - 66048*n^5 + 446400*n^4 - 1620808*n^3 + 3339890*n^2 - 3711613*n + 1743218)*a(n) = 128*(536870912*n^13 - 14831058944*n^12 + 188986949632*n^11 - 1471608258560*n^10 + 7817645654016*n^9 - 29941451735040*n^8 + 85134250240000*n^7 - 182149348773632*n^6 + 293626158621632*n^5 - 352753169299376*n^4 + 307548490429492*n^3 - 184675145918224*n^2 + 68635535585133*n - 11961900200250)*a(n-1) + 16777216*(4096*n^6 - 41472*n^5 + 177600*n^4 - 413768*n^3 + 556826*n^2 - 414321*n + 135135)*a(n-2). - Vaclav Kotesovec, Dec 07 2014
a(n) ~ 8^(8*(n-1)-1/2) / 7^(7*(n-1)-1/2) * n^(n-2) / exp(n-1). - Vaclav Kotesovec, Dec 07 2014

A251579 E.g.f.: exp(9xG(x)^8) / G(x)^8 where G(x) = 1 + x*G(x)^9 is the g.f. of A062994.

Original entry on oeis.org

1, 1, 9, 225, 10017, 656289, 57255849, 6262226721, 825067217025, 127305462542913, 22527254639457801, 4498536675388410081, 1000890043482114644769, 245556248365681036646625, 65862976584851401437170217, 19174678419336874098038167329, 6022064808176665662053835550209

Offset: 0

Paul D. Hanna, Dec 06 2014

nonn

			E.g.f.: A(x) = 1 + x + 9*x^2/2! + 225*x^3/3! + 10017*x^4/4! + 656289*x^5/5! +...
such that A(x) = exp(9*x*G(x)^8) / G(x)^8
where G(x) = 1 + x*G(x)^9 is the g.f. of A062994:
G(x) = 1 + x + 9*x^2 + 117*x^3 + 1785*x^4 + 29799*x^5 + 527085*x^6 +...
Note that
A'(x) = exp(9*x*G(x)^8) = 1 + 9*x + 225*x^2/2! + 10017*x^3/3! +...
LOGARITHMIC DERIVATIVE.
The logarithm of the e.g.f. begins:
log(A(x)) = x + 8*x^2/2 + 200*x^3/3 + 8976*x^4/4 + 592368*x^5/5 +...
and so A'(x)/A(x) = G(x)^8.
TABLE OF POWERS OF E.G.F.
Form a table of coefficients of x^k/k! in A(x)^n as follows.
n=1: [1, 1,   9,  225,  10017,   656289,  57255849,  6262226721, ...];
n=2: [1, 2,  20,  504,  22320,  1453248, 126104256, 13731880320, ...];
n=3: [1, 3,  33,  843,  37233,  2411667, 208241361, 22581193851, ...];
n=4: [1, 4,  48, 1248,  55104,  3554496, 305558784, 33002857728, ...];
n=5: [1, 5,  65, 1725,  76305,  4906965, 420159825, 45211985325, ...];
n=6: [1, 6,  84, 2280, 101232,  6496704, 554376384, 59448214656, ...];
n=7: [1, 7, 105, 2919, 130305,  8353863, 710786601, 75977951175, ...];
n=8: [1, 8, 128, 3648, 163968, 10511232, 892233216, 95096756736, ...]; ...
in which the main diagonal begins (see A251587):
[1, 2, 33, 1248, 76305, 6496704, 710786601, 95096756736, ...]
and is given by the formula:
[x^n/n!] A(x)^(n+1) = 9^(n-7) * (n+1)^(n-8) * (262144*n^7 + 2494464*n^6 + 10470208*n^5 + 25229505*n^4 + 37857568*n^3 + 35537670*n^2 + 19414368*n + 4782969) for n>=0.

Cf. A251589, A251669, A062994, A234513.

Cf. Variants: A243953, A251573, A251574, A251575, A251576, A251577, A251578, A251580.

Flatten[{1,1,Table[Sum[9^k * n!/k! * Binomial[9*n-k-9, n-k] * (k-1)/(n-1),{k,0,n}],{n,2,20}]}] (* Vaclav Kotesovec, Dec 07 2014 *)

{a(n) = local(G=1);for(i=1,n,G=1+x*G^9 +x*O(x^n)); n!*polcoeff(exp(9*x*G^8)/G^8, n)}
for(n=0, 20, print1(a(n), ", "))

{a(n) = if(n==0|n==1, 1, sum(k=0, n, 9^k * n!/k! * binomial(9*n-k-9,n-k) * (k-1)/(n-1) ))}
for(n=0, 20, print1(a(n), ", "))

Let G(x) = 1 + x*G(x)^9 be the g.f. of A062994, then the e.g.f. A(x) of this sequence satisfies:
(1) A'(x)/A(x) = G(x)^8.
(2) A'(x) = exp(8*x*G(x)^8).
(3) A(x) = exp( Integral G(x)^8 dx ).
(4) A(x) = exp( Sum_{n>=1} A234513(n-1)*x^n/n ), where A234513(n-1) = binomial(9*n-2,n)/(8*n-1).
(5) A(x) = F(x/A(x)) where F(x) is the e.g.f. of A251589.
(6) A(x) = Sum_{n>=0} A251589(n)*(x/A(x))^n/n! and
(7) [x^n/n!] A(x)^(n+1) = (n+1)*A251589(n),
where A251589(n) = 9^(n-7) * (n+1)^(n-9) * (262144*n^7 + 2494464*n^6 + 10470208*n^5 + 25229505*n^4 + 37857568*n^3 + 35537670*n^2 + 19414368*n + 4782969).
a(n) = Sum_{k=0..n} 9^k * n!/k! * binomial(9*n-k-9, n-k) * (k-1)/(n-1) for n>1.
Recurrence: 128*(2*n-3)*(4*n-7)*(4*n-5)*(8*n-15)*(8*n-13)*(8*n-11)*(8*n-9)*(59049*n^7 - 1102248*n^6 + 8858079*n^5 - 39764115*n^4 + 107806473*n^3 - 176772075*n^2 + 162618742*n - 64907105)*a(n) = 81*(282429536481*n^15 - 8943601988565*n^14 + 132044525265870*n^13 - 1206188364304287*n^12 + 7627178203628841*n^11 - 35382975568258428*n^10 + 124478964551078775*n^9 - 338415281830783431*n^8 + 717436315214480025*n^7 - 1187215577095780764*n^6 + 1522794566607803919*n^5 - 1488866286016780047*n^4 + 1075889068341959448*n^3 - 543536112365518695*n^2 + 172059320987344825*n - 25799292366848000)*a(n-1) - 387420489*(59049*n^7 - 688905*n^6 + 3484620*n^5 - 9940725*n^4 + 17352558*n^3 - 18650247*n^2 + 11527801*n - 3203200)*a(n-2). - Vaclav Kotesovec, Dec 07 2014
a(n) ~ 9^(9*(n-1)-1/2) / 8^(8*(n-1)-1/2) * n^(n-2) / exp(n-1). - Vaclav Kotesovec, Dec 07 2014

A251580 E.g.f.: exp(10xG(x)^9) / G(x)^9 where G(x) = 1 + x*G(x)^10 is the g.f. of A059968.

Original entry on oeis.org

1, 1, 10, 280, 13960, 1023760, 99935200, 12226859200, 1801725932800, 310890328768000, 61516405597830400, 13735605457885312000, 3416919943285809280000, 937247149729410729472000, 281051240591439955878400000, 91474949907165746668607488000, 32117399444469103248129863680000

Offset: 0

Paul D. Hanna, Dec 06 2014

nonn

In general, Sum_{k=0..n} m^k * n!/k! * binomial(m*n-k-m, n-k) * (k-1)/(n-1) is for m>1 asymptotic to m^(m*(n-1)-1/2) / (m-1)^((m-1)*(n-1)-1/2) * n^(n-2) / exp(n-1). - Vaclav Kotesovec, Dec 07 2014

			E.g.f.: A(x) = 1 + x + 10*x^2/2! + 280*x^3/3! + 13960*x^4/4! + 1023760*x^5/5! +...
such that A(x) = exp(10*x*G(x)^9) / G(x)^9
where G(x) = 1 + x*G(x)^10 is the g.f. of A059968:
G(x) = 1 + x + 10*x^2 + 145*x^3 + 2470*x^4 + 46060*x^5 + 910252*x^6 +...
Note that
A'(x) = exp(10*x*G(x)^9) = 1 + 10*x + 280*x^2/2! + 13960*x^3/3! +...
LOGARITHMIC DERIVATIVE.
The logarithm of the e.g.f. begins:
log(A(x)) = x + 9*x^2/2 + 252*x^3/3 + 12654*x^4/4 + 933984*x^5/5 +...
and so A'(x)/A(x) = G(x)^9.
TABLE OF POWERS OF E.G.F.
Form a table of coefficients of x^k/k! in A(x)^n as follows.
n=1: [1, 1,  10,  280,  13960,  1023760,   99935200,  12226859200, ...];
n=2: [1, 2,  22,  620,  30760,  2243120,  217911520,  26556406400, ...];
n=3: [1, 3,  36, 1026,  50760,  3683880,  356283360,  43256151360, ...];
n=4: [1, 4,  52, 1504,  74344,  5374240,  517647520,  62621962240, ...];
n=5: [1, 5,  70, 2060, 101920,  7344920,  704861200,  84980501600, ...];
n=6: [1, 6,  90, 2700, 133920,  9629280,  921060720, 110691813600, ...];
n=7: [1, 7, 112, 3430, 170800, 12263440, 1169680960, 140152067440, ...];
n=8: [1, 8, 136, 4256, 213040, 15286400, 1454475520, 173796462080, ...]; ...
in which the main diagonal begins (see A251587):
[1, 2, 36, 1504, 101920, 9629280, 1169680960, 173796462080, ...]
and is given by the formula:
[x^n/n!] A(x)^(n+1) = 10^(n-8) * (n+1)^(n-9) * (4782969*n^8 + 50309748*n^7 + 237013938*n^6 + 655232760*n^5 + 1166624361*n^4 + 1374998212*n^3 + 1051760172*n^2 + 479277840*n + 100000000) for n>=0.

Cf. A251590, A251670, A059968, A234573.

Cf. Variants: A243953, A251573, A251574, A251575, A251576, A251577, A251578, A251579.

Flatten[{1,1,Table[Sum[10^k * n!/k! * Binomial[10*n-k-10, n-k] * (k-1)/(n-1),{k,0,n}],{n,2,20}]}] (* Vaclav Kotesovec, Dec 07 2014 *)

{a(n) = local(G=1);for(i=1,n, G = 1 + x*G^10 +x*O(x^n)); n!*polcoeff( exp(10*x*G^9) / G^9, n)}
for(n=0, 20, print1(a(n), ", "))

{a(n) = if(n==0, 1, sum(k=0, n, 10^k * n!/k! * binomial(10*n-k-10,n-k)*if(n==1,1/10,(k-1)/(n-1)) ))}
for(n=0, 20, print1(a(n), ", "))

Let G(x) = 1 + x*G(x)^10 be the g.f. of A059968, then the e.g.f. A(x) of this sequence satisfies:
(1) A'(x)/A(x) = G(x)^9.
(2) A'(x) = exp(10*x*G(x)^9).
(3) A(x) = exp( Integral G(x)^9 dx ).
(4) A(x) = exp( Sum_{n>=1} A234573(n-1)*x^n/n ), where A234573(n-1) = binomial(10*n-2,n)/(9*n-1).
(5) A(x) = F(x/A(x)) where F(x) is the e.g.f. of A251590.
(6) A(x) = Sum_{n>=0} A251590(n)*(x/A(x))^n/n! and
(7) [x^n/n!] A(x)^(n+1) = (n+1)*A251590(n),
where A251590(n) = 10^(n-8) * (n+1)^(n-10) * (4782969*n^8 + 50309748*n^7 + 237013938*n^6 + 655232760*n^5 + 1166624361*n^4 + 1374998212*n^3 + 1051760172*n^2 + 479277840*n + 100000000).
a(n) = Sum_{k=0..n} 10^k * n!/k! * binomial(10*n-k-10, n-k) * (k-1)/(n-1) for n>1.
Recurrence: 81*(3*n-5)*(3*n-4)*(9*n-17)*(9*n-16)*(9*n-14)*(9*n-13)*(9*n-11)*(9*n-10)*(250000*n^8 - 5300000*n^7 + 49332500*n^6 - 263500000*n^5 + 884055975*n^4 - 1909634570*n^3 + 2596659373*n^2 - 2035277286*n + 705468040)*a(n) = 800*(3125000000000*n^17 - 111562500000000*n^16 + 1872125000000000*n^15 - 19618187500000000*n^14 + 143829395937500000*n^13 - 783195370343750000*n^12 + 3281447638218750000*n^11 - 10810863753751875000*n^10 + 28370066880833218750*n^9 - 59681174371832246875*n^8 + 100725400409628775000*n^7 - 135736802338370325750*n^6 + 144424061701272600950*n^5 - 118936947986511839915*n^4 + 73322264536912326596*n^3 - 31942069342168467356*n^2 + 8798129066413437408*n - 1156512281566561920)*a(n-1) + 10000000000*(250000*n^8 - 3300000*n^7 + 19232500*n^6 - 64805000*n^5 + 138543475*n^4 - 193260670*n^3 + 172779013*n^2 - 91243350*n + 22054032)*a(n-2). - Vaclav Kotesovec, Dec 07 2014
a(n) ~ 10^(10*(n-1)-1/2) / 9^(9*(n-1)-1/2) * n^(n-2) / exp(n-1). - Vaclav Kotesovec, Dec 07 2014

A251663 E.g.f.: exp( 3xG(x)^2 ) / G(x), where G(x) = 1 + x*G(x)^3 is the g.f. A001764.

Original entry on oeis.org

1, 2, 11, 120, 2061, 48918, 1487151, 55188108, 2419385625, 122367255498, 7014349322739, 449405251066368, 31826192109186789, 2468711973793223070, 208159999898813165079, 18957203713618483723092, 1854424578467714146269489, 193922780991931737971748882, 21588348501840566333913576795

Offset: 0

Paul D. Hanna, Dec 07 2014

nonn

			E.g.f.: A(x) = 1 + 2*x + 11*x^2/2! + 120*x^3/3! + 2061*x^4/4! + 48918*x^5/5! +...
such that A(x) = exp(3*x*G(x)^2) / G(x)
where G(x) = 1 + x*G(x)^3 is the g.f. of A001764:
G(x) = 1 + x + 3*x^2 + 12*x^3 + 55*x^4 + 273*x^5 + 1428*x^6 +...
The e.g.f. satisfies:
A(x) = 1 + 2*x/A(x)^2 + 27*x^2/(2!*A(x)^4) + 756*x^3/(3!*A(x)^6) + 32805*x^4/(4!*A(x)^8) + 1940598*x^5/(5!*A(x)^10) + 145746783*x^6/(6!*A(x)^12) + 13286025000*x^7/(7!*A(x)^14) +...+ (n+1)*(2*n+1)^(n-2)*3^n * x^n/(n!*A(x)^(2*n)) +...

Cf. A251573, A251693, A001764.

Cf. Variants: A243953, A251664, A251665, A251666, A251667, A251668, A251669, A251670.

Table[Sum[3^k * n!/k! * Binomial[3*n-k-2, n-k] * (2*k-1)/(2*n-1),{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Dec 07 2014 *)

{a(n)=local(G=1); for(i=0, n, G=1+x*G^3 +x*O(x^n)); n!*polcoeff(exp(3*x*G^2)/G, n)}
for(n=0, 20, print1(a(n), ", "))

{a(n) = sum(k=0, n, 3^k * n!/k! * binomial(3*n-k-2,n-k) * (2*k-1)/(2*n-1) )}
for(n=0, 20, print1(a(n), ", "))

Let G(x) = 1 + x*G(x)^3 be the g.f. of A001764, then the e.g.f. A(x) of this sequence satisfies:
(1) A'(x)/A(x) = G(x)^2 + G'(x)/G(x).
(2) A(x) = F(x/A(x)^2) where F(x) is the e.g.f. of A251693.
(3) A(x) = Sum_{n>=0} A251693(n)*(x/A(x)^2)^n/n! where A251693(n) = (n+1) * (2*n+1)^(n-2) * 3^n.
(4) [x^n/n!] A(x)^(2*n+1) = (n+1) * (2*n+1)^(n-1) * 3^n.
a(n) = Sum_{k=0..n} 3^k * n!/k! * binomial(3*n-k-2, n-k) * (2*k-1)/(2*n-1) for n>=0.
Recurrence: 2*(2*n-1)*(9*n^2 - 30*n + 19)*a(n) = 3*(81*n^4 - 432*n^3 + 756*n^2 - 393*n - 88)*a(n-1) - 27*(9*n^2 - 12*n - 2)*a(n-2). - Vaclav Kotesovec, Dec 07 2014
a(n) ~ 3^(3*n-3/2) * n^(n-1) / (2^(2*n-1/2) * exp(n-1)). - Vaclav Kotesovec, Dec 07 2014

cp's OEIS Frontend

A127670 Discriminants of Chebyshev S-polynomials A049310.

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A251573 E.g.f.: exp(3*x*G(x)^2) / G(x)^2 where G(x) = 1 + x*G(x)^3 is the g.f. of A001764.

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A251574 E.g.f.: exp(4*x*G(x)^3) / G(x)^3 where G(x) = 1 + x*G(x)^4 is the g.f. of A002293.

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A251575 E.g.f.: exp(5*x*G(x)^4) / G(x)^4 where G(x) = 1 + x*G(x)^5 is the g.f. of A002294.

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A251576 E.g.f.: exp(6*x*G(x)^5) / G(x)^5 where G(x) = 1 + x*G(x)^6 is the g.f. of A002295.

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A251577 E.g.f.: exp(7*x*G(x)^6) / G(x)^6 where G(x) = 1 + x*G(x)^7 is the g.f. of A002296.

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A251578 E.g.f.: exp(8*x*G(x)^7) / G(x)^7 where G(x) = 1 + x*G(x)^8 is the g.f. of A007556.

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A251573 E.g.f.: exp(3xG(x)^2) / G(x)^2 where G(x) = 1 + x*G(x)^3 is the g.f. of A001764.

A251574 E.g.f.: exp(4xG(x)^3) / G(x)^3 where G(x) = 1 + x*G(x)^4 is the g.f. of A002293.

A251575 E.g.f.: exp(5xG(x)^4) / G(x)^4 where G(x) = 1 + x*G(x)^5 is the g.f. of A002294.

A251576 E.g.f.: exp(6xG(x)^5) / G(x)^5 where G(x) = 1 + x*G(x)^6 is the g.f. of A002295.

A251577 E.g.f.: exp(7xG(x)^6) / G(x)^6 where G(x) = 1 + x*G(x)^7 is the g.f. of A002296.

A251578 E.g.f.: exp(8xG(x)^7) / G(x)^7 where G(x) = 1 + x*G(x)^8 is the g.f. of A007556.

A251579 E.g.f.: exp(9xG(x)^8) / G(x)^8 where G(x) = 1 + x*G(x)^9 is the g.f. of A062994.

A251580 E.g.f.: exp(10xG(x)^9) / G(x)^9 where G(x) = 1 + x*G(x)^10 is the g.f. of A059968.

A251663 E.g.f.: exp( 3xG(x)^2 ) / G(x), where G(x) = 1 + x*G(x)^3 is the g.f. A001764.