A251660 Table of coefficients in functions R(n,x) defined by R(n,x) = exp( n*x*G(n,x)^(n-1) ) / G(n,x)^(n-1) where G(n,x) = 1 + x*G(n,x)^n, for rows n>=1.

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 8, 1, 1, 1, 4, 21, 56, 1, 1, 1, 5, 40, 261, 592, 1, 1, 1, 6, 65, 712, 4833, 8512, 1, 1, 1, 7, 96, 1505, 18784, 120303, 155584, 1, 1, 1, 8, 133, 2736, 51505, 663424, 3778029, 3456896, 1, 1, 1, 9, 176, 4501, 115056, 2354725, 29480896, 143531433, 90501632, 1

Offset

1,9

Links

Table of n, a(n) for n=1..66.

Formula

E.g.f. of row n, R(n,x), for n>=1, satisfies:

(1) [x^k/k!] R(n,x)^(k+1) = n^(k-1) * (n+k) * (k+1)^(k-2) for k>=0.

(2) R(n,x) = exp( n*x*G(n,x)^(n-1) ) / G(n,x)^(n-1), where G(n,x) = 1 + x*G(n,x)^n.

(3) R'(n,x)/R(n,x) = G(n,x)^(n-1), where G(n,x) = 1 + x*G(n,x)^n.

T(n,k) = Sum_{j=0..k} n^j * k!/j! * binomial(n*(k-1)-j, k-j) * (j-1)/(k-1) for k>1, n>=1.

Example

This table begins:
n=1: [1, 1,  1,   1,     1,       1,        1,           1, ...];
n=2: [1, 1,  2,   8,    56,     592,     8512,      155584, ...];
n=3: [1, 1,  3,  21,   261,    4833,   120303,     3778029, ...];
n=4: [1, 1,  4,  40,   712,   18784,   663424,    29480896, ...];
n=5: [1, 1,  5,  65,  1505,   51505,  2354725,   135258625, ...];
n=6: [1, 1,  6,  96,  2736,  115056,  6455376,   454666176, ...];
n=7: [1, 1,  7, 133,  4501,  224497, 14926387,  1245099709, ...];
n=8: [1, 1,  8, 176,  6896,  397888, 30584128,  2948178304, ...];
n=9: [1, 1,  9, 225, 10017,  656289, 57255849,  6262226721, ...];
n=10:[1, 1, 10, 280, 13960, 1023760, 99935200, 12226859200, ...]; ...
where e.g.f. of row n equals: exp( n*x*G(n,x)^(n-1) ) / G(n,x)^(n-1).
Related table of coefficients in G(n,x) = 1 + x*G(n,x)^n  begins:
n=1: [1, 1,  1,   1,    1,     1,      1,        1, ...];
n=2: [1, 1,  2,   5,   14,    42,    132,      429, ...];
n=3: [1, 1,  3,  12,   55,   273,   1428,     7752, ...];
n=4: [1, 1,  4,  22,  140,   969,   7084,    53820, ...];
n=5: [1, 1,  5,  35,  285,  2530,  23751,   231880, ...];
n=6: [1, 1,  6,  51,  506,  5481,  62832,   749398, ...];
n=7: [1, 1,  7,  70,  819, 10472, 141778 , 1997688, ...];
n=8: [1, 1,  8,  92, 1240, 18278, 285384,  4638348, ...];
n=9: [1, 1,  9, 117, 1785, 29799, 527085,  9706503, ...];
n=10:[1, 1, 10, 145, 2470, 46060, 910252, 18730855, ...]; ...

Prog

(PARI) {T(n, k)=local(G=1); for(i=0, k, G=1+x*G^n +x*O(x^k)); k!*polcoeff(exp(n*x*G^(n-1))/G^(n-1), k)}
/* Print as a rectangular table */
for(n=1, 10, for(k=0, 10, print1(T(n, k), ", ")); print(""))
/* Print as a flattened table */
for(n=0, 12, for(k=0, n, print1(T(n-k+1, k), ", ")); )
/* Print the Related table of functions G(n, x) = 1 + x*G(n, x)^n */
{R(n, k)=local(G=1); for(i=0, k, G=1+x*G^n +x*O(x^k)); polcoeff(G, k)}
for(n=1, 10, for(k=0, 10, print1(R(n, k), ", ")); print(""))
(PARI) /* Binomial sum formula for term T(n, k) */
{T(n, k) = if(k<=1, 1, sum(j=0, k, n^j * k!/j! * binomial(n*(k-1)-j, k-j) * (j-1)/(k-1)))}
for(n=1, 10, for(k=0, 10, print1(T(n, k), ", ")); print(""))

Crossrefs

Cf. Rows: A243953, A251573, A251574, A251575, A251576, A251577, A251578, A251579, A251580.

Sequence in context: A112707 A196017 A343555 * A279453 A054252 A240472

Adjacent sequences: A251657 A251658 A251659 * A251661 A251662 A251663

Keyword

nonn,tabl

Author

Paul D. Hanna, Dec 21 2014

Status

approved