A340940 Rectangular table of coefficients T(k,n) in row functions R(k,x) = sum_{n>=0} T(k,n)*x^n that satisfy the condition: Sum_{n>=0} x^n/(1 - x*R(k,x)^(n+k)) = Sum_{n>=0} x^n*R(k,x)^n/(1 - x*R(k,x)^(k*n+k-1)), for k >= 0, read here by antidiagonals.

1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 3, 9, 10, 1, 1, 4, 18, 46, 36, 1, 1, 5, 30, 126, 253, 137, 1, 1, 6, 45, 266, 958, 1467, 545, 1, 1, 7, 63, 482, 2578, 7707, 8842, 2244, 1, 1, 8, 84, 790, 5665, 26501, 64564, 54878, 9500, 1, 1, 9, 108, 1206, 10896, 70725, 284016, 557519, 348489, 41151

Offset

0,9

Links

Table of n, a(n) for n=0..65.

Formula

Given g.f. of row k, R(k,x), the following sums are all equal:

(1) B(k,x) = Sum_{n>=0} x^n / (1 - x*R(k,x)^(n+k)),

(2) B(k,x) = Sum_{n>=0} x^n*R(k,x)^n / (1 - x*R(k,x)^(k*n+k-1)),

(3) B(k,x) = Sum_{n>=0} x^n*R(k,x)^((k-1)*n) / (1 - x*R(k,x)^(k*n+1)),

(4) B(k,x) = Sum_{n>=0} x^n*R(k,x)^(k*n) / (1 - x*R(k,x)^n),

(5) B(k,x) = Sum_{n>=0} x^(2*n) * R(k,x)^(n^2+k*n) * (1 - x^2*R(k,x)^(2*n+k)) / ((1 - x*R(k,x)^n)*(1 - x*R(k,x)^(n+k))),

(6) B(k,x) = Sum_{n>=0} x^(2*n) * R(k,x)^(k*n^2+k*n) * (1 - x^2*R(k,x)^(2*k*n+k)) / ((1 - x*R(k,x)^(k*n+1))*(1 - x*R(k,x)^(k*n+k-1)));

see the example section for the coefficients in B(k,x).

Example

This table of coefficients in row functions R(k,x) = sum_{n>=0} T(k,n)*x^n begins:
k=0: 1, 1, 1, 3, 10, 36, 137, 545, 2244, 9500, ...;
k=1: 1, 1, 2, 9, 46, 253, 1467, 8842, 54878, 348489, ...;
k=2: 1, 1, 3, 18, 126, 958, 7707, 64564, 557519, 4928784, ...;
k=3: 1, 1, 4, 30, 266, 2578, 26501, 284016, 3139627, 35546887, ...;
k=4: 1, 1, 5, 45, 482, 5665, 70725, 921174, 12379878, 170435921, ...;
k=5: 1, 1, 6, 63, 790, 10896, 159783, 2445499, 38627339, 625074945, ...;
k=6: 1, 1, 7, 84, 1206, 19073, 320903, 5636558, 102186707, 1898039195, ...;
k=7: 1, 1, 8, 108, 1746, 31123, 590433, 11695452, 239129756, 5009457416, ...;
k=8: 1, 1, 9, 135, 2426, 48098, 1015137, 22373051, 508995136, 11864425419, ...;
k=9: 1, 1, 10, 165, 3262, 71175, 1653491, 40115036, 1004638668, 25778507966, ...; ...
where the row functions R(k,x) satisfy the condition:
B(k,x) = Sum_{n>=0} x^n/(1 - x*R(k,x)^(n+k)) ) and
B(k,x) = Sum_{n>=0} x^n*R(k,x)^n/(1 - x*R(k,x)^(k*m+k-1)) are equal.
Note that this condition is satisfied by every row function in the table; however, rows k=0 and k=1 require special handling to determine the coefficients; see A340941 (row 0) and A340942 (row 1) for further information.
RELATED SERIES.
Corresponding to each row function R(k,x), we have the series
B(k,x) = Sum_{n>=0} x^n / (1 - x*R(k,x)^(n+k));
the table of coefficients in B(k,x) begins:
k=0: 1, 2, 3, 5, 10, 25, 75, 255, 940, 3660, 14827, ...;
k=1: 1, 2, 4, 10, 34, 148, 749, 4138, 24160, 146556, ...;
k=2: 1, 2, 5, 18, 93, 602, 4406, 34666, 286098, 2443548, ...;
k=3: 1, 2, 6, 29, 203, 1738, 16574, 168779, 1797190, 19770290, ...;
k=4: 1, 2, 7, 43, 380, 4032, 47234, 588683, 7657593, 102796547, ...;
k=5: 1, 2, 8, 60, 640, 8085, 112116, 1649968, 25311223, 400396030, ...;
k=6: 1, 2, 9, 80, 999, 14623, 233995, 3966533, 70020324, 1273840402, ...;
k=7: 1, 2, 10, 103, 1473, 24497, 443987, 8512318, 169822712, 3489867908, ...;
k=8: 1, 2, 11, 129, 2078, 38683, 782845, 16739843, 372233838, 8522500051, ...;
k=9: 1, 2, 12, 158, 2830, 58282, 1302255, 30715554, 752955814, 18998262705, ...; ...

Prog

(PARI) {T(k, n) = my(A=[1, 1], H);
if(k>=2,
for(i=1, n, A=concat(A, 0); H=A; A=concat(A, 0);
H[#A-1] = -polcoeff( sum(m=0, #A, x^m/(1 - x*Ser(A)^(m+k)) ) - sum(m=0, #A, x^m*Ser(A)^m/(1 - x*Ser(A)^(k*m+k-1)) ), #A)/(k-1); A=H); A[n+1],
if(k==0,
A=1+x +x^3*O(x^n); H=A;
for(k=1, n, A = (A-x)*(1-x*A) * sum(m=0, n+3, x^m / (1 - x*A^m +x^3*O(x^n)) );
A = truncate( H + polcoeff(A, k+2)*x^k ) +x^3*O(x^n); H=A); polcoeff(A, n),
if(k==1, if(n<2, 1,
Vec(-(x-1)^n*Ser(vector(n+1, j, T(j+1, n))))[n])
)))}
/* Print rectangular table: */
for(k=0, 10, for(n=0, 10, print1(T(k, n), ", ")); print(""))
/* As read by antidiagonals: */
for(k=0, 10, for(n=0, k, print1(T(k, k-n), ", ")); )

Crossrefs

Cf. A340941 (row 0), A340942 (row 1), A340894 (row 2), A340895 (row 3), A340943 (row 4), A341376 (row 5).

Sequence in context: A071947 A139343 A059247 * A362464 A244665 A194518

Adjacent sequences: A340937 A340938 A340939 * A340941 A340942 A340943

Keyword

nonn,tabl

Author

Paul D. Hanna, Feb 03 2021

Status

approved