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Fisk, Polynomials, roots, and interlacing, Thm 1.43)","lean_route":null,"lean_declarations":[],"certificate_hash":"a6c199596485c5de2c9da5985929f5fe1484f35a8182047fb30964b48cb0f2fb","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-c77d5d4d44ca1f43","role":"sequence","name":"Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 3*x + 2.","uses":{"claim":"real_rooted","from_index":0,"region":{}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)"},{"position":1,"node_id":"n-c77d5d4d44ca1f43","role":"sequence","name":"Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 3*x + 2.","uses":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"uses_text":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1"}]}],"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:    3,        2;\n  1:    6,       22,          4;\n  2:   12,      152,        136,           8;\n  3:   24,      880,       2244,         756,           16;\n  4:   48,     4712,      27236,       25872,         3980,           32;\n  5:   96,    24328,     281612,      577852,       260452,        20396,           64;\n  6:  192,   123464,    2654580,    10223568,      9927328,      2418456,       103164,         128;\n  7:  384,   621544,   23644828,   156817644,    280454360,    149544896,     21349836,      518572,        256;\n  8:  768,  3117320,  203144836,  2190696368,   6563248468,   6439611752,   2064796300,   182416544,    2599132,      512;\n  9: 1536, 15608104, 1704653676, 28706251068, 135257965500, 220236273180, 131009050116, 26844599604, 1525354452, 13009740, 1024;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2+(3*n-4)*x)*P(n-2,x) + (3*x-3*x^2)*d/dx P(n-2,x) + (1+(3*n+2)*x)*P(n-1,x) + (3*x-3*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 3 + 2*x, P(1,x) = 6 + 22*x + 4*x^2.","mathematica":"P[0, x_] := 3 + 2*x;\nP[1, x_] := 6 + 22*x + 4*x^2;\nP[n_, x_] := P[n, x] = Expand[(2 + (3*n - 4)*x)*P[n - 2, x] + (3*x - 3*x^2)*D[P[n - 2, x], x] + (1 + (3*n + 2)*x)*P[n - 1, x] + (3*x - 3*x^2)*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 3 + 2*x; v[2] = 6 + 22*x + 4*x^2; for(n = 2, N, my(i = n + 1); v[i] = (2 + (3*n - 4)*x)*v[i - 2] + (3*x - 3*x^2)*deriv(v[i - 2], x) + (1 + (3*n + 2)*x)*v[i - 1] + (3*x - 3*x^2)*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 - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-c77d5d4d44ca1f43","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-31517a5bce5d126f"],"text":"A257610 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A257610 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}