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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:       11,         24;\n  3:      106,        432,        432;\n  4:     1011,       5904,      11232,        6912;\n  5:     9726,      73440,     203040,      241920,      103680;\n  6:    94526,     873360,    3153600,     5529600,     4665600,     1492992;\n  7:   926836,   10118304,   45027360,   104025600,   130636800,    83607552,    20901888;\n  8:  9153731,  115235232,  609118272,  1745210880,  2910297600,  2800852992,  1421328384,   286654464;\n  9: 90941686, 1296575424, 7934697792, 27129779712, 56422033920, 72613158912, 55933452288, 23219011584, 3869835264;\n","recurrence_sentence":"The row polynomials satisfy (2*n^2-3*n)*P(n,x) = -(32*n^2-80*n+32+(192*n^2-480*n+192)*x+(288*n^2-720*n+288)*x^2)*P(n-2,x) + (24*n^2-48*n+22+(48*n^2-96*n+48)*x)*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[(-(32*n^2 - 80*n + 32 + (192*n^2 - 480*n + 192)*x + (288*n^2 - 720*n + 288)*x^2)*P[n - 2, x] + (24*n^2 - 48*n + 22 + (48*n^2 - 96*n + 48)*x)*P[n - 1, 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] = (-(32*n^2 - 80*n + 32 + (192*n^2 - 480*n + 192)*x + (288*n^2 - 720*n + 288)*x^2)*v[i - 2] + (24*n^2 - 48*n + 22 + (48*n^2 - 96*n + 48)*x)*v[i - 1])/(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 A013611","basis A013610"],"edges":["e-a70bef035c61e98a","e-57859db25abb5763"],"text":"A367270 \u2192 basis A013611 \u2192 basis A013610"},"headline":"A367270 \u2192 basis A013611 \u2192 basis A013610","chain":null}