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T(n, k) = (1/2) * binomial(2*(n - k + 1), n - k + 1) * binomial(2*n - k, k - 1) for n > 0, T(0, 0) = 1.","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:      1;\n  1:      1,       1;\n  2:      5,       9,        2;\n  3:     25,      67,       30,        3;\n  4:    129,     485,      340,       70,        4;\n  5:    681,    3491,     3490,     1135,      135,       5;\n  6:   3653,   25095,    34076,    15785,     2975,     231,      6;\n  7:  19825,  180391,   322854,   201537,    53970,    6650,    364,     7;\n  8: 108545, 1297413,  2998892,  2436798,   873978,  152670,  13272,   540,   8;\n  9: 598417, 9339259, 27475950, 28382475, 13136403, 3059133, 376320, 24330, 765, 9;\n","recurrence_sentence":"The row polynomials satisfy (2*n^2-3*n)*P(n,x) = -(2*n^2-5*n+2+(6*n^2-15*n+6)*x+(2*n^2-5*n+2)*x^2)*P(n-2,x) - ((6*n^2-15*n+6)*x+(4*n^2-10*n+4)*x^2)*d/dx P(n-2,x) - (2*n^2-5*n+2)*x^2*d^2/dx^2 P(n-2,x) + (12*n^2-24*n+10+(4*n^2-8*n+4)*x)*P(n-1,x) + (4*n^2-8*n+4)*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(2*n^2 - 5*n + 2 + (6*n^2 - 15*n + 6)*x + (2*n^2 - 5*n + 2)*x^2)*P[n - 2, x] - ((6*n^2 - 15*n + 6)*x + (4*n^2 - 10*n + 4)*x^2)*D[P[n - 2, x], x] - (2*n^2 - 5*n + 2)*x^2*D[P[n - 2, x], {x, 2}] + (12*n^2 - 24*n + 10 + (4*n^2 - 8*n + 4)*x)*P[n - 1, x] + (4*n^2 - 8*n + 4)*x*D[P[n - 1, x], x])/(2*n^2 - 3*n)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + x; for(n = 2, N, my(i = n + 1); v[i] = (-(2*n^2 - 5*n + 2 + (6*n^2 - 15*n + 6)*x + (2*n^2 - 5*n + 2)*x^2)*v[i - 2] - ((6*n^2 - 15*n + 6)*x + (4*n^2 - 10*n + 4)*x^2)*deriv(v[i - 2], x) - (2*n^2 - 5*n + 2)*x^2*deriv(deriv(v[i - 2], x), x) + (12*n^2 - 24*n + 10 + (4*n^2 - 8*n + 4)*x)*v[i - 1] + (4*n^2 - 8*n + 4)*x*deriv(v[i - 1], x))/(2*n^2 - 3*n)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-0a5e54eacdb24124","steps":["basis A008277"],"edges":["e-74401799fd385997"],"text":"A360282 \u2192 basis A008277"},"headline":"A360282 \u2192 basis A008277","chain":null}