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 1-8 of 8 results.

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.

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

Views

Author

Paul D. Hanna, Dec 06 2014

Keywords

Examples

			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.

Links

Crossrefs

Cf. A251583, A251663, A001764, A006013.
Cf. Variants: A243953, A251574, A251575, A251576, A251577, A251578, A251579, A251580.

Programs

  • Mathematica
    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 *)
  • PARI
    {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), ", "))
  • PARI
    {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), ", "))

Formula

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

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).

Original entry on oeis.org

1, 1, 9, 202, 7745, 429192, 31383169, 2862173104, 313456891041, 40120056928000, 5880757402778489, 971776774765633536, 178783183332534538849, 36248462186643418851328, 8031317081954231731640625, 1930762515644331053063077888, 500587184017640118192794723009, 139240118930461640299714951839744
Offset: 0

Views

Author

Paul D. Hanna, Dec 06 2014

Keywords

Examples

			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.

Links

Crossrefs

Cf. A251577, A002296.
Cf. Variants: A251583, A251584, A251585, A251586, A251588, A251589, A251590.

Programs

  • Mathematica
    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 *)
  • PARI
    {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),", "))
  • PARI
    {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), ", "))

Formula

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

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

Views

Author

Paul D. Hanna, Dec 06 2014

Keywords

Examples

			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.

Links

Crossrefs

Cf. A251574, A001764.
Cf. Variants: A251583, A251585, A251586, A251587, A251588, A251589, A251590.

Programs

  • Magma
    [4^(n - 2)*(n + 1)^(n - 4)*(3*n^2 + 13*n + 16): n in [0..50]]; // G. C. Greubel, Nov 13 2017
  • Mathematica
    Table[4^(n - 2)*(n + 1)^(n - 4)*(3*n^2 + 13*n + 16), {n, 0, 50}] (* G. C. Greubel, Nov 13 2017 *)
  • PARI
    {a(n) = 4^(n-2) * (n+1)^(n-4) * (3*n^2 + 13*n + 16) }
for(n=0,20,print1(a(n),", "))
  • PARI
    {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), ", "))

Formula

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) * (16*n^3 + 87*n^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

Views

Author

Paul D. Hanna, Dec 06 2014

Keywords

Examples

			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.

Links

Crossrefs

Cf. A251575, A002294.
Cf. Variants: A251583, A251584, A251586, A251587, A251588, A251589, A251590.

Programs

  • Magma
    [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
  • Mathematica
    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 *)
  • PARI
    {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),", "))
  • PARI
    {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), ", "))

Formula

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) * (125*n^4 + 810*n^3 + 2095*n^2 + 2586*n + 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

Views

Author

Paul D. Hanna, Dec 06 2014

Keywords

Examples

			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.

Links

Crossrefs

Cf. A251576, A002295.
Cf. Variants: A251583, A251584, A251585, A251587, A251588, A251589, A251590.

Programs

  • Magma
    [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
  • Mathematica
    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 *)
  • PARI
    {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),", "))
  • PARI
    {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), ", "))

Formula

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) * (16807*n^6 + 143031*n^5 + 525875*n^4 + 1074745*n^3 + 1294846*n^2 + 876856*n + 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

Views

Author

Paul D. Hanna, Dec 06 2014

Keywords

Examples

			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.

Links

Crossrefs

Cf. A251578, A007556.
Cf. Variants: A251583, A251584, A251585, A251586, A251587, A251589, A251590.

Programs

  • Magma
    [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
  • Mathematica
    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 *)
  • PARI
    {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),", "))
  • PARI
    {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), ", "))

Formula

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

A251589 a(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).

Original entry on oeis.org

1, 1, 11, 312, 15261, 1082784, 101540943, 11887094592, 1671909996537, 274908280855680, 51777320665313331, 10995461917161191424, 2599903208658745278549, 677539978514149068122112, 192961347474468607479984375, 59630505640075441195821563904, 19874186375795055056211917494257
Offset: 0

Views

Author

Paul D. Hanna, Dec 06 2014

Keywords

Examples

			E.g.f.: A(x) = 1 + x + 11*x^2/2! + 312*x^3/3! + 15261*x^4/4! + 1082784*x^5/5! +...
such that A(x) = exp( 9*x*A(x) * G(x*A(x))^8 ) / G(x*A(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 +...
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 + 9*x^2/2! + 225*x^3/3! + 10017*x^4/4! + 656289*x^5/5! +...
is the e.g.f. of A251579.

Links

Crossrefs

Cf. A251579, A062994.
Cf. Variants: A251583, A251584, A251585, A251586, A251587, A251588, A251590.

Programs

  • Magma
    [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): n in [0..10]]; // G. C. Greubel, Nov 13 2017
  • Mathematica
    Table[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), {n, 0, 50}] (* G. C. Greubel, Nov 13 2017 *)
  • PARI
    {a(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)}
for(n=0,20,print1(a(n),", "))
  • PARI
    {a(n) = local(G=1,A=1); for(i=1,n, G=1+x*G^9 +x*O(x^n));
for(i=1,n, A = exp(9*x*A * subst(G^8,x,x*A) ) / subst(G^8,x,x*A) ); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))

Formula

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) = exp( 9*x*A(x) * G(x*A(x))^8 ) / G(x*A(x))^8.
(2) A(x) = F(x*A(x)) where F(x) = exp(9*x*G(x)^8)/G(x)^8 is the e.g.f. of A251579.
(3) a(n) = [x^n/n!] F(x)^(n+1)/(n+1) where F(x) is the e.g.f. of A251579.
E.g.f.: -LambertW(-9*x) * (9 + LambertW(-9*x))^8 / (x*9^9). - Vaclav Kotesovec, Dec 07 2014

A251590 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).

Original entry on oeis.org

1, 1, 12, 376, 20384, 1604880, 167097280, 21724557760, 3393929393280, 619917062849920, 129708290461760000, 30601444681382400000, 8038933665661600000000, 2327544788978773504000000, 736481767572932400000000000, 252867957872989831168000000000, 93638020040848371127040000000000
Offset: 0

Views

Author

Paul D. Hanna, Dec 06 2014

Keywords

Examples

			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.

Links

Crossrefs

Cf. A251580, A059968.
Cf. Variants: A251583, A251584, A251585, A251586, A251587, A251588, A251589.

Programs

  • Magma
    [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
  • Mathematica
    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 *)
  • PARI
    {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),", "))
  • PARI
    {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), ", "))

Formula

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
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