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Triangle read by rows, T(n, k) for 0 <= k <= n.","uses":{"claim":"real_rooted","from_index":1,"region":{}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)"},{"position":1,"node_id":"n-e9164c616dcc1bbd","role":"sequence","name":"Coefficients of a family of orthogonal polynomials. Triangle read by rows, T(n, k) for 0 <= k <= n.","uses":{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"},"uses_text":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2"}]}],"used_by":[],"used_by_total":0,"links":[],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:       2,         1;\n   2:       3,         7,          1;\n   3:       8,        44,         16,          1;\n   4:      30,       310,        203,         29,         1;\n   5:     144,      2472,       2528,        608,        46,         1;\n   6:     840,     22120,      32620,      11620,      1435,        67,        1;\n   7:    5760,    219840,     443840,     217960,     38840,      2908,       92,      1;\n   8:   45360,   2404080,    6403320,    4138120,    993370,    105658,     5299,    121,     1;\n   9:  403200,  28694400,   98040320,   80682560,  24961440,   3526320,   248640,   8928,   154,   1;\n  10: 3991680, 371226240, 1591349760, 1627077760, 628878320, 113016624, 10483032, 525288, 14163, 191, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (n-2+x)*P(n-2,x) + 3*x*d/dx P(n-2,x) + (n-1+x)*P(n-1,x) + 3*x*d/dx P(n-1,x) for n >= 3, with P(1,x) = 2 + x, P(2,x) = 3 + 7*x + x^2.","mathematica":"P[1, x_] := 2 + x;\nP[2, x_] := 3 + 7*x + x^2;\nP[n_, x_] := P[n, x] = Expand[(n - 2 + x)*P[n - 2, x] + 3*x*D[P[n - 2, x], x] + (n - 1 + x)*P[n - 1, x] + 3*x*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 2 + x; v[2] = 3 + 7*x + x^2; for(n = 3, N, my(i = n); v[i] = (n - 2 + x)*v[i - 2] + 3*x*deriv(v[i - 2], x) + (n - 1 + x)*v[i - 1] + 3*x*deriv(v[i - 1], x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-e9164c616dcc1bbd","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-11c2cb91bf9b2413"],"text":"A322944 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A322944 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}