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T(n, k) = binomial(n, k)*binomial(n - 1, n - k - 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;\n  2:     3,      2;\n  3:    10,     15,       3;\n  4:    35,     94,      42,       4;\n  5:   126,    555,     425,      90,      5;\n  6:   462,   3191,    3765,    1340,    165,      6;\n  7:  1716,  18109,   31087,   16590,   3395,    273,     7;\n  8:  6435, 102110,  246162,  185108,  55930,   7420,   420,   8;\n  9: 24310, 574107, 1898157, 1934562, 808899, 156870, 14574, 612, 9;\n","recurrence_sentence":"The row polynomials satisfy (2*n^2-3*n)*P(n,x) = -((2*n^2-5*n+2)*x+(2*n^2-5*n+2)*x^2)*P(n-2,x) - ((2*n^2-5*n+2)*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) + (8*n^2-16*n+6+(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.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-((2*n^2 - 5*n + 2)*x + (2*n^2 - 5*n + 2)*x^2)*P[n - 2, x] - ((2*n^2 - 5*n + 2)*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}] + (8*n^2 - 16*n + 6 + (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; for(n = 2, N, my(i = n + 1); v[i] = (-((2*n^2 - 5*n + 2)*x + (2*n^2 - 5*n + 2)*x^2)*v[i - 2] - ((2*n^2 - 5*n + 2)*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) + (8*n^2 - 16*n + 6 + (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-b1b19325e0094118","steps":["basis A008277"],"edges":["e-f7f7b35ffa4336f4"],"text":"A367270 \u2192 basis A008277"},"headline":"A367270 \u2192 basis A008277","chain":null}