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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. 6
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 (list; table; graph; refs; listen; history; text; internal format)
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 * A244665 A194518 A023572

Adjacent sequences:  A340937 A340938 A340939 * A340941 A340942 A340943

KEYWORD

nonn,tabl

AUTHOR

Paul D. Hanna, Feb 03 2021

STATUS

approved

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Last modified August 2 02:22 EDT 2021. Contains 346408 sequences. (Running on oeis4.)