|
%I #17 Jun 18 2021 16:38:49
%S 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,
%T 1,6,45,266,958,1467,545,1,1,7,63,482,2578,7707,8842,2244,1,1,8,84,
%U 790,5665,26501,64564,54878,9500,1,1,9,108,1206,10896,70725,284016,557519,348489,41151
%N 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.
%F Given g.f. of row k, R(k,x), the following sums are all equal:
%F (1) B(k,x) = Sum_{n>=0} x^n / (1 - x*R(k,x)^(n+k)),
%F (2) B(k,x) = Sum_{n>=0} x^n*R(k,x)^n / (1 - x*R(k,x)^(k*n+k-1)),
%F (3) B(k,x) = Sum_{n>=0} x^n*R(k,x)^((k-1)*n) / (1 - x*R(k,x)^(k*n+1)),
%F (4) B(k,x) = Sum_{n>=0} x^n*R(k,x)^(k*n) / (1 - x*R(k,x)^n),
%F (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))),
%F (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)));
%F see the example section for the coefficients in B(k,x).
%e This table of coefficients in row functions R(k,x) = sum_{n>=0} T(k,n)*x^n begins:
%e k=0: 1, 1, 1, 3, 10, 36, 137, 545, 2244, 9500, ...;
%e k=1: 1, 1, 2, 9, 46, 253, 1467, 8842, 54878, 348489, ...;
%e k=2: 1, 1, 3, 18, 126, 958, 7707, 64564, 557519, 4928784, ...;
%e k=3: 1, 1, 4, 30, 266, 2578, 26501, 284016, 3139627, 35546887, ...;
%e k=4: 1, 1, 5, 45, 482, 5665, 70725, 921174, 12379878, 170435921, ...;
%e k=5: 1, 1, 6, 63, 790, 10896, 159783, 2445499, 38627339, 625074945, ...;
%e k=6: 1, 1, 7, 84, 1206, 19073, 320903, 5636558, 102186707, 1898039195, ...;
%e k=7: 1, 1, 8, 108, 1746, 31123, 590433, 11695452, 239129756, 5009457416, ...;
%e k=8: 1, 1, 9, 135, 2426, 48098, 1015137, 22373051, 508995136, 11864425419, ...;
%e k=9: 1, 1, 10, 165, 3262, 71175, 1653491, 40115036, 1004638668, 25778507966, ...; ...
%e where the row functions R(k,x) satisfy the condition:
%e B(k,x) = Sum_{n>=0} x^n/(1 - x*R(k,x)^(n+k)) ) and
%e B(k,x) = Sum_{n>=0} x^n*R(k,x)^n/(1 - x*R(k,x)^(k*m+k-1)) are equal.
%e 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.
%e RELATED SERIES.
%e Corresponding to each row function R(k,x), we have the series
%e B(k,x) = Sum_{n>=0} x^n / (1 - x*R(k,x)^(n+k));
%e the table of coefficients in B(k,x) begins:
%e k=0: 1, 2, 3, 5, 10, 25, 75, 255, 940, 3660, 14827, ...;
%e k=1: 1, 2, 4, 10, 34, 148, 749, 4138, 24160, 146556, ...;
%e k=2: 1, 2, 5, 18, 93, 602, 4406, 34666, 286098, 2443548, ...;
%e k=3: 1, 2, 6, 29, 203, 1738, 16574, 168779, 1797190, 19770290, ...;
%e k=4: 1, 2, 7, 43, 380, 4032, 47234, 588683, 7657593, 102796547, ...;
%e k=5: 1, 2, 8, 60, 640, 8085, 112116, 1649968, 25311223, 400396030, ...;
%e k=6: 1, 2, 9, 80, 999, 14623, 233995, 3966533, 70020324, 1273840402, ...;
%e k=7: 1, 2, 10, 103, 1473, 24497, 443987, 8512318, 169822712, 3489867908, ...;
%e k=8: 1, 2, 11, 129, 2078, 38683, 782845, 16739843, 372233838, 8522500051, ...;
%e k=9: 1, 2, 12, 158, 2830, 58282, 1302255, 30715554, 752955814, 18998262705, ...; ...
%o (PARI) {T(k,n) = my(A=[1,1],H);
%o if(k>=2,
%o for(i=1,n, A=concat(A,0); H=A; A=concat(A,0);
%o 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],
%o if(k==0,
%o A=1+x +x^3*O(x^n);H=A;
%o 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)) );
%o A = truncate( H + polcoeff(A, k+2)*x^k ) +x^3*O(x^n); H=A);polcoeff(A,n),
%o if(k==1, if(n<2,1,
%o Vec(-(x-1)^n*Ser(vector(n+1,j,T(j+1,n))))[n])
%o )))}
%o /* Print rectangular table: */
%o for(k=0,10, for(n=0, 10, print1(T(k,n),", "));print(""))
%o /* As read by antidiagonals: */
%o for(k=0,10, for(n=0,k, print1(T(k,k-n),", "));)
%Y Cf. A340941 (row 0), A340942 (row 1), A340894 (row 2), A340895 (row 3), A340943 (row 4), A341376 (row 5).
%K nonn,tabl
%O 0,9
%A _Paul D. Hanna_, Feb 03 2021
|
