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A168646
Triangle read by rows (0 <= k <= n): T(n,k) = [x^k] p(x,n), where p(x,0) = 1, p(x,n) = (8 - n)*(x + 1)^n - (7 - n)*(x^n + 1) for 1 <= n <= 6, and p(x,n) = 6*(x + 1)^n - Sum_{i=0..4} (Sum_{j=0..i} binomial(n, j)*(x^j + x^(n - j))) for n >= 7.
3
1, 1, 1, 1, 12, 1, 1, 15, 15, 1, 1, 16, 24, 16, 1, 1, 15, 30, 30, 15, 1, 1, 12, 30, 40, 30, 12, 1, 1, 14, 63, 105, 105, 63, 14, 1, 1, 16, 84, 224, 280, 224, 84, 16, 1, 1, 18, 108, 336, 630, 630, 336, 108, 18, 1, 1, 20, 135, 480, 1050, 1512, 1050, 480, 135, 20, 1
OFFSET
0,5
EXAMPLE
Triangle begins:
1;
1, 1;
1, 12, 1;
1, 15, 15, 1;
1, 16, 24, 16, 1;
1, 15, 30, 30, 15, 1;
1, 12, 30, 40, 30, 12, 1;
1, 14, 63, 105, 105, 63, 14, 1;
1, 16, 84, 224, 280, 224, 84, 16, 1;
1, 18, 108, 336, 630, 630, 336, 108, 18, 1;
1, 20, 135, 480, 1050, 1512, 1050, 480, 135, 20, 1;
...
MATHEMATICA
p[x_, n_] := If[n == 0, 1, If[n == 1, x + 1, 6*(x + 1)^n - (x^n + 1) - If[n > 2, (x^n + n*x^(n - 1) + n*x + 1), (x^n + 1)] - If[ n > 3, (x^n + n*x^( n - 1) + Binomial[n, n - 2]*x^(n - 2) + Binomial[n, n - 2]*x^2 + n*x + 1), (x^n + 1)] - If[n > 4, (x^n + n*x^( n - 1) + Binomial[n, n - 2]*x^(n - 2) + Binomial[n, n - 3]*x^(n - 3) + Binomial[ n, n - 3]*x^3 + Binomial[n, n - 2]*x^2 + n*x + 1), (x^n + 1)] - If[n > 5, (x^n + n*x^(n - 1) + Binomial[n, n - 2]*x^( n - 2) + Binomial[n, n - 3]*x^(n - 3) + Binomial[n, n - 4]*x^( n - 4) + Binomial[n, n - 4]*x^4 + Binomial[n, n - 3]*x^3 + Binomial[n, n - 2]*x^2 + n*x + 1), (x^n + 1)]]];
Flatten[Table[CoefficientList[p[x, n], x], {n, 0, 10}]]
PROG
(Maxima) T(n, k) := if k = 0 or k = n then 1 else (if n <= 6 then (8 - n)*binomial(n, k) else ratcoef(6*(x + 1)^n - sum(sum(binomial(n, j)*(x^j + x^(n - j)), j, 1, i), i, 1, 4), x, k))$
create_list(T(n, k), n, 0, 12, k, 0, n); /* Franck Maminirina Ramaharo, Jan 02 2019 */
CROSSREFS
KEYWORD
nonn,tabl,easy,less
AUTHOR
EXTENSIONS
Edited by Franck Maminirina Ramaharo, Jan 02 2019
STATUS
approved