A251589 -id:A251589 - cp's OEIS frontend

A251587 a(n) = 7^(n-5) * (n+1)^(n-7) * (1296n^5 + 9720n^4 + 30555n^3 + 50665n^2 + 44621*n + 16807).

1, 1, 9, 202, 7745, 429192, 31383169, 2862173104, 313456891041, 40120056928000, 5880757402778489, 971776774765633536, 178783183332534538849, 36248462186643418851328, 8031317081954231731640625, 1930762515644331053063077888, 500587184017640118192794723009, 139240118930461640299714951839744

Offset: 0

Paul D. Hanna, Dec 06 2014

			E.g.f.: A(x) = 1 + x + 9*x^2/2! + 202*x^3/3! + 7745*x^4/4! + 429192*x^5/5! +...
such that A(x) = exp( 7*x*A(x) * G(x*A(x))^6 ) / G(x*A(x))^6
where G(x) = 1 + x*G(x)^5 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 +...
RELATED SERIES.
Note that A(x) = F(x*A(x)) where F(x) = exp(7*x*G(x)^6)/G(x)^6,
F(x) = 1 + x + 7*x^2/2! + 133*x^3/3! + 4501*x^4/4! + 224497*x^5/5! +...
is the e.g.f. of A251577.

G. C. Greubel, Table of n, a(n) for n = 0..300

Cf. A251577, A002296.

Cf. Variants: A251583, A251584, A251585, A251586, A251588, A251589, A251590.

Table[7^(n - 5)*(n + 1)^(n - 7)*(1296*n^5 + 9720*n^4 + 30555*n^3 + 50665*n^2 + 44621*n + 16807), {n, 0, 20}] (* G. C. Greubel, Nov 10 2017 *)

{a(n) = 7^(n-5) * (n+1)^(n-7) * (1296*n^5 + 9720*n^4 + 30555*n^3 + 50665*n^2 + 44621*n + 16807)}
for(n=0,20,print1(a(n),", "))

{a(n) = local(G=1,A=1); for(i=1,n, G=1+x*G^7 +x*O(x^n));
for(i=1,n, A = exp(7*x*A * subst(G^6,x,x*A) ) / subst(G^6,x,x*A) ); n!*polcoeff(A, n)}
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) = exp( 7*x*A(x) * G(x*A(x))^6 ) / G(x*A(x))^6.
(2) A(x) = F(x*A(x)) where F(x) = exp(7*x*G(x)^6)/G(x)^6 is the e.g.f. of A251577.
(3) a(n) = [x^n/n!] F(x)^(n+1)/(n+1) where F(x) is the e.g.f. of A251577.
E.g.f.: -LambertW(-7*x) * (7 + LambertW(-7*x))^6 / (x*7^7). - 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

A251583 a(n) = 3^(n-1) * (n+1)^(n-3) * (n+3).

Original entry on oeis.org

1, 1, 5, 54, 945, 23328, 750141, 29859840, 1420541793, 78732000000, 4986357828309, 355459848339456, 28178328756432465, 2459548529521606656, 234438580086767578125, 24233149581890213117952, 2700277512299794365456321, 322689729227525728790446080, 41170357602396483760424637477, 5585797616762880000000000000000

Offset: 0

Paul D. Hanna, Dec 06 2014

nonn

			E.g.f.: A(x) = 1 + x + 5*x^2/2! + 54*x^3/3! + 945*x^4/4! + 23328*x^5/5! +...
such that A(x) = exp( 3*x*A(x) * G(x*A(x))^2 ) / G(x*A(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 +...
RELATED SERIES.
Note that A(x) = F(x*A(x)) where F(x) = exp(3*x*G(x)^2)/G(x)^2,
F(x) = 1 + x + 3*x^2/2! + 21*x^3/3! + 261*x^4/4! + 4833*x^5/5! +...
is the e.g.f. of A251573.

G. C. Greubel, Table of n, a(n) for n = 0..330

Cf. A251573, A001764.

Cf. Variants: A251584, A251585, A251586, A251587, A251588, A251589, A251590.

Table[3^(n - 1)*(n + 1)^(n - 3)*(n + 3), {n, 0, 20}] (* G. C. Greubel, Nov 12 2017 *)

{a(n) = 3^(n-1) * (n+1)^(n-3) * (n+3)}
for(n=0,20,print1(a(n),", "))

{a(n) = local(G=1,A=1); for(i=1,n, G=1+x*G^3 +x*O(x^n));
for(i=1,n, A = exp(3*x*A * subst(G^2,x,x*A) ) / subst(G^2,x,x*A) ); n!*polcoeff(A, n)}
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) = exp( 3*x*A(x) * G(x*A(x))^2 ) / G(x*A(x))^2.
(2) A(x) = F(x*A(x)) where F(x) = exp(3*x*G(x)^2)/G(x)^2 is the e.g.f. of A251573.
(3) a(n) = [x^n/n!] F(x)^(n+1)/(n+1) where F(x) is the e.g.f. of A251573.
E.g.f.: -LambertW(-3*x) * (3 + LambertW(-3*x))^2 / (27*x). - Vaclav Kotesovec, Dec 07 2014

A251584 a(n) = 4^(n-2) * (n+1)^(n-4) * (3n^2 + 13n + 16).

Original entry on oeis.org

1, 1, 6, 82, 1856, 59904, 2533888, 133169152, 8384643072, 616038400000, 51781055676416, 4903194794655744, 516634816527990784, 59967828129860288512, 7604226293760000000000, 1046004175955626414833664, 155145294199098982239567872, 24683771056652857103433596928, 4193546668531304927540807729152, 757730637305085952000000000000000

Offset: 0

Paul D. Hanna, Dec 06 2014

nonn

			E.g.f.: A(x) = 1 + x + 6*x^2/2! + 82*x^3/3! + 1856*x^4/4! + 59904*x^5/5! + ...
such that A(x) = exp( 4*x*A(x) * G(x*A(x))^3 ) / G(x*A(x))^3
where G(x) = 1 + x*G(x)^3 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 + ...
RELATED SERIES.
Note that A(x) = F(x*A(x)) where F(x) = exp(4*x*G(x)^3)/G(x)^3,
F(x) = 1 + x + 4*x^2/2! + 40*x^3/3! + 712*x^4/4! + 18784*x^5/5! + ...
is the e.g.f. of A251574.

G. C. Greubel, Table of n, a(n) for n = 0..322

Cf. A251574, A001764.

Cf. Variants: A251583, A251585, A251586, A251587, A251588, A251589, A251590.

[4^(n - 2)*(n + 1)^(n - 4)*(3*n^2 + 13*n + 16): n in [0..50]]; // G. C. Greubel, Nov 13 2017

Table[4^(n - 2)*(n + 1)^(n - 4)*(3*n^2 + 13*n + 16), {n, 0, 50}] (* G. C. Greubel, Nov 13 2017 *)

{a(n) = 4^(n-2) * (n+1)^(n-4) * (3*n^2 + 13*n + 16) }
for(n=0,20,print1(a(n),", "))

{a(n) = local(G=1,A=1); for(i=1,n, G=1+x*G^4 +x*O(x^n));for(i=1,n, A = exp(4*x*A * subst(G^3,x,x*A) ) / subst(G^3,x,x*A) ); n!*polcoeff(A, n)}
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) = exp( 4*x*A(x) * G(x*A(x))^3 ) / G(x*A(x))^3.
(2) A(x) = F(x*A(x)) where F(x) = exp(4*x*G(x)^3)/G(x)^3 is the e.g.f. of A251574.
(3) a(n) = [x^n/n!] F(x)^(n+1)/(n+1) where F(x) is the e.g.f. of A251574.
E.g.f.: -LambertW(-4*x) * (4 + LambertW(-4*x))^3 / (256*x). - Vaclav Kotesovec, Dec 07 2014

A251585 a(n) = 5^(n-3) * (n+1)^(n-5) * (16n^3 + 87n^2 + 172*n + 125).

Original entry on oeis.org

1, 1, 7, 116, 3229, 129000, 6776875, 443200000, 34766465625, 3185000000000, 333992093359375, 39470976000000000, 5192072114658203125, 752537122540000000000, 119176291179656982421875, 20476256583680000000000000, 3793880513498167242431640625, 754086862404270000000000000000

Offset: 0

Paul D. Hanna, Dec 06 2014

nonn

			E.g.f.: A(x) = 1 + x + 7*x^2/2! + 116*x^3/3! + 3229*x^4/4! + 129000*x^5/5! + ...
such that A(x) = exp( 5*x*A(x) * G(x*A(x))^4 ) / G(x*A(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 + ...
RELATED SERIES.
Note that A(x) = F(x*A(x)) where F(x) = exp(5*x*G(x)^4)/G(x)^4,
F(x) = 1 + x + 5*x^2/2! + 65*x^3/3! + 1505*x^4/4! + 51505*x^5/5! + ...
is the e.g.f. of A251575.

G. C. Greubel, Table of n, a(n) for n = 0..312

Cf. A251575, A002294.

Cf. Variants: A251583, A251584, A251586, A251587, A251588, A251589, A251590.

[5^(n - 3)*(n + 1)^(n - 5)*(16*n^3 + 87*n^2 + 172*n + 125): n in [0..50]]; // G. C. Greubel, Nov 13 2017

Table[5^(n - 3)*(n + 1)^(n - 5)*(16*n^3 + 87*n^2 + 172*n + 125), {n, 0, 50}] (* G. C. Greubel, Nov 13 2017 *)

{a(n) = 5^(n-3) * (n+1)^(n-5) * (16*n^3 + 87*n^2 + 172*n + 125)}
for(n=0,20,print1(a(n),", "))

{a(n) = local(G=1,A=1); for(i=1,n, G=1+x*G^5 +x*O(x^n));
for(i=1,n, A = exp(5*x*A * subst(G^4,x,x*A) ) / subst(G^4,x,x*A) ); n!*polcoeff(A, n)}
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) = exp( 5*x*A(x) * G(x*A(x))^4 ) / G(x*A(x))^4.
(2) A(x) = F(x*A(x)) where F(x) = exp(5*x*G(x)^4)/G(x)^4 is the e.g.f. of A251575.
(3) a(n) = [x^n/n!] F(x)^(n+1)/(n+1) where F(x) is the e.g.f. of A251575.
E.g.f.: -LambertW(-5*x) * (5 + LambertW(-5*x))^4 / (x*5^5). - Vaclav Kotesovec, Dec 07 2014

A251586 a(n) = 6^(n-4) * (n+1)^(n-6) * (125n^4 + 810n^3 + 2095n^2 + 2586n + 1296).

Original entry on oeis.org

1, 1, 8, 156, 5160, 245976, 15450912, 1209613824, 113666333184, 12479546880000, 1568823886181376, 222308476014034944, 35069155573323036672, 6096327654732137496576, 1158040133351856000000000, 238674982804212474577944576, 53050036437721656891731017728, 12649916782354997981599305302016

Offset: 0

Paul D. Hanna, Dec 06 2014

nonn

			E.g.f.: A(x) = 1 + x + 8*x^2/2! + 156*x^3/3! + 5160*x^4/4! + 245976*x^5/5! +...
such that A(x) = exp( 6*x*A(x) * G(x*A(x))^5 ) / G(x*A(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 +...
RELATED SERIES.
Note that A(x) = F(x*A(x)) where F(x) = exp(6*x*G(x)^5)/G(x)^5,
F(x) = 1 + x + 6*x^2/2! + 96*x^3/3! + 2736*x^4/4! + 115056*x^5/5! +...
is the e.g.f. of A251576.

G. C. Greubel, Table of n, a(n) for n = 0..306

Cf. A251576, A002295.

Cf. Variants: A251583, A251584, A251585, A251587, A251588, A251589, A251590.

[6^(n - 4)*(n + 1)^(n - 6)*(125*n^4 + 810*n^3 + 2095*n^2 + 2586*n + 1296): n in [0..50]]; // G. C. Greubel, Nov 13 2017

Table[6^(n - 4)*(n + 1)^(n - 6)*(125*n^4 + 810*n^3 + 2095*n^2 + 2586*n + 1296), {n, 0, 50}] (* G. C. Greubel, Nov 13 2017 *)

{a(n) = 6^(n-4) * (n+1)^(n-6) * (125*n^4 + 810*n^3 + 2095*n^2 + 2586*n + 1296)}
for(n=0,20,print1(a(n),", "))

{a(n) = local(G=1,A=1); for(i=1,n, G=1+x*G^6 +x*O(x^n));
for(i=1,n, A = exp(6*x*A * subst(G^5,x,x*A) ) / subst(G^5,x,x*A) ); n!*polcoeff(A, n)}
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) = exp( 6*x*A(x) * G(x*A(x))^5 ) / G(x*A(x))^5.
(2) A(x) = F(x*A(x)) where F(x) = exp(6*x*G(x)^5)/G(x)^5 is the e.g.f. of A251576.
(3) a(n) = [x^n/n!] F(x)^(n+1)/(n+1) where F(x) is the e.g.f. of A251576.
E.g.f.: -LambertW(-6*x) * (6 + LambertW(-6*x))^5 / (x*6^6). - Vaclav Kotesovec, Dec 07 2014

A251588 a(n) = 8^(n-6) * (n+1)^(n-8) * (16807n^6 + 143031n^5 + 525875n^4 + 1074745n^3 + 1294846n^2 + 876856n + 262144).

Original entry on oeis.org

1, 1, 10, 254, 11080, 700008, 58411696, 6082359760, 760774053888, 111229735731200, 18626295180427264, 3516652429787529216, 739238816214490808320, 171262175332556483854336, 43359709355122360320000000, 11911510903698787868252045312, 3529104034183977458725537447936, 1121766516051874786454563454976000

Offset: 0

Paul D. Hanna, Dec 06 2014

nonn

			E.g.f.: A(x) = 1 + x + 10*x^2/2! + 254*x^3/3! + 11080*x^4/4! + 700008*x^5/5! +...
such that A(x) = exp( 7*x*A(x) * G(x*A(x))^6 ) / G(x*A(x))^6
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 +...
RELATED SERIES.
Note that A(x) = F(x*A(x)) where F(x) = exp(8*x*G(x)^7)/G(x)^7,
F(x) = 1 + x + 8*x^2/2! + 176*x^3/3! + 6896*x^4/4! + 397888*x^5/5! +...
is the e.g.f. of A251578.

G. C. Greubel, Table of n, a(n) for n = 0..296

Cf. A251578, A007556.

Cf. Variants: A251583, A251584, A251585, A251586, A251587, A251589, A251590.

[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): n in [0..50]]; // G. C. Greubel, Nov 13 2017

Table[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), {n, 0, 50}] (* G. C. Greubel, Nov 13 2017 *)

{a(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)}
for(n=0,20,print1(a(n),", "))

{a(n) = local(G=1,A=1); for(i=1,n, G=1+x*G^8 +x*O(x^n));
for(i=1,n, A = exp(8*x*A * subst(G^7,x,x*A) ) / subst(G^7,x,x*A) ); n!*polcoeff(A, n)}
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) = exp( 8*x*A(x) * G(x*A(x))^7 ) / G(x*A(x))^7.
(2) A(x) = F(x*A(x)) where F(x) = exp(8*x*G(x)^7)/G(x)^7 is the e.g.f. of A251578.
(3) a(n) = [x^n/n!] F(x)^(n+1)/(n+1) where F(x) is the e.g.f. of A251578.
E.g.f.: -LambertW(-8*x) * (8 + LambertW(-8*x))^7 / (x*8^8). - Vaclav Kotesovec, Dec 07 2014

A251590 a(n) = 10^(n-8) * (n+1)^(n-10) * (4782969n^8 + 50309748n^7 + 237013938n^6 + 655232760n^5 + 1166624361n^4 + 1374998212n^3 + 1051760172n^2 + 479277840n + 100000000).

Original entry on oeis.org

1, 1, 12, 376, 20384, 1604880, 167097280, 21724557760, 3393929393280, 619917062849920, 129708290461760000, 30601444681382400000, 8038933665661600000000, 2327544788978773504000000, 736481767572932400000000000, 252867957872989831168000000000, 93638020040848371127040000000000

Offset: 0

Paul D. Hanna, Dec 06 2014

nonn

			E.g.f.: A(x) = 1 + x + 12*x^2/2! + 376*x^3/3! + 20384*x^4/4! + 1604880*x^5/5! +...
such that A(x) = exp( 10*x*A(x) * G(x*A(x))^9 ) / G(x*A(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 +...
RELATED SERIES.
Note that A(x) = F(x*A(x)) where F(x) = exp(9*x*G(x)^8)/G(x)^8,
F(x) = 1 + x + 10*x^2/2! + 280*x^3/3! + 13960*x^4/4! + 1023760*x^5/5! +...
is the e.g.f. of A251580.

G. C. Greubel, Table of n, a(n) for n = 0..289

Cf. A251580, A059968.

Cf. Variants: A251583, A251584, A251585, A251586, A251587, A251588, A251589.

[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): n in [0..50]]; // G. C. Greubel, Nov 13 2017

Table[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), {n, 0, 50}] (* G. C. Greubel, Nov 13 2017 *)

{a(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)}
for(n=0,20,print1(a(n),", "))

{a(n) = local(G=1,A=1); for(i=1,n, G=1+x*G^10 +x*O(x^n));
for(i=1,n, A = exp(10*x*A * subst(G^9,x,x*A) ) / subst(G^9,x,x*A) ); n!*polcoeff(A, n)}
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) = exp( 10*x*A(x) * G(x*A(x))^9 ) / G(x*A(x))^9.
(2) A(x) = F(x*A(x)) where F(x) = exp(10*x*G(x)^9)/G(x)^9 is the e.g.f. of A251580.
(3) a(n) = [x^n/n!] F(x)^(n+1)/(n+1) where F(x) is the e.g.f. of A251580.
E.g.f.: -LambertW(-10*x) * (10 + LambertW(-10*x))^9 / (x*10^10). - Vaclav Kotesovec, Dec 07 2014

cp's OEIS Frontend

A251587 a(n) = 7^(n-5) * (n+1)^(n-7) * (1296*n^5 + 9720*n^4 + 30555*n^3 + 50665*n^2 + 44621*n + 16807).

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

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A251583 a(n) = 3^(n-1) * (n+1)^(n-3) * (n+3).

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A251584 a(n) = 4^(n-2) * (n+1)^(n-4) * (3*n^2 + 13*n + 16).

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A251585 a(n) = 5^(n-3) * (n+1)^(n-5) * (16*n^3 + 87*n^2 + 172*n + 125).

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A251586 a(n) = 6^(n-4) * (n+1)^(n-6) * (125*n^4 + 810*n^3 + 2095*n^2 + 2586*n + 1296).

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A251588 a(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).

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A251587 a(n) = 7^(n-5) * (n+1)^(n-7) * (1296n^5 + 9720n^4 + 30555n^3 + 50665n^2 + 44621*n + 16807).

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.

A251584 a(n) = 4^(n-2) * (n+1)^(n-4) * (3n^2 + 13n + 16).

A251585 a(n) = 5^(n-3) * (n+1)^(n-5) * (16n^3 + 87n^2 + 172*n + 125).

A251586 a(n) = 6^(n-4) * (n+1)^(n-6) * (125n^4 + 810n^3 + 2095n^2 + 2586n + 1296).

A251588 a(n) = 8^(n-6) * (n+1)^(n-8) * (16807n^6 + 143031n^5 + 525875n^4 + 1074745n^3 + 1294846n^2 + 876856n + 262144).

A251590 a(n) = 10^(n-8) * (n+1)^(n-10) * (4782969n^8 + 50309748n^7 + 237013938n^6 + 655232760n^5 + 1166624361n^4 + 1374998212n^3 + 1051760172n^2 + 479277840n + 100000000).