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A355259 Triangle read by rows. Row k are the coefficients of the polynomials (sorted by ascending powers) which interpolate the points (n, A355257(n, k+1)) for n = 0..k.

%I A355259 #5 Jul 04 2022 06:51:09
%S A355259 1,3,2,14,12,3,90,82,30,4,744,680,285,60,5,7560,6788,2985,760,105,6,
%T A355259 91440,80136,35532,9870,1715,168,7,1285200,1098984,482300,138796,
%U A355259 27160,3444,252,8,20603520,17227584,7425492,2152584,447405,65520,6342,360,9
%N A355259 Triangle read by rows. Row k are the coefficients of the polynomials (sorted by ascending powers) which interpolate the points (n, A355257(n, k+1)) for n = 0..k.
%C A355259 Conjecture from _Mélika Tebni_: These polynomials generate column k + 1 of
%C A355259 A355257.
%e A355259 [0]        1;
%e A355259 [1]        3,        2;
%e A355259 [2]       14,       12,       3;
%e A355259 [3]       90,       82,      30,       4;
%e A355259 [4]      744,      680,     285,      60,      5;
%e A355259 [5]     7560,     6788,    2985,     760,    105,     6;
%e A355259 [6]    91440,    80136,   35532,    9870,   1715,   168,    7;
%e A355259 [7]  1285200,  1098984,  482300,  138796,  27160,  3444,  252,   8;
%e A355259 [8] 20603520, 17227584, 7425492, 2152584, 447405, 65520, 6342, 360, 9;
%e A355259 .
%e A355259 Seen as polynomials:
%e A355259 p0(x) = 1;
%e A355259 p1(x) = 3 + 2*x;
%e A355259 p2(x) = 14 + 12*x + 3*x^2;
%e A355259 p3(x) = 90 + 82*x + 30*x^2 + 4*x^3;
%e A355259 p4(x) = 744 + 680*x + 285*x^2 + 60*x^3 + 5*x^4;
%e A355259 p5(x) = 7560 + 6788*x + 2985*x^2 + 760*x^3 + 105*x^4 + 6*x^5;
%e A355259 p6(x) = 91440 + 80136*x + 35532*x^2 + 9870*x^3 + 1715*x^4 + 168*x^5 + 7*x^6;
%p A355259 A355257 := (n, k) -> add(k!*binomial(k + n - 1, k - j - 1)/(j + 1), j = 0..k-1):
%p A355259 for k from 0 to 9 do CurveFitting:-PolynomialInterpolation([seq([n, A355257(n, k+1)], n = 0..k)], x):
%p A355259 print(seq(coeff(%, x, j), j = 0..k)) od:
%Y A355259 Cf. A355257.
%K A355259 nonn,tabl
%O A355259 0,2
%A A355259 _Peter Luschny_, Jul 03 2022