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the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"225c9777c2495b0efb59407597f67a545563f261b97fd37a5e0843553689fd46","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-e655e253465c7aff","role":"sequence","name":"veronese parent of Binomial coefficients C(n,k) with n-k odd, read by rows.","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:    2,     4,      1;\n  1:    4,     5,      1;\n  2:    8,    46,     46,      13,       1;\n  3:   16,    65,     55,      14,       1;\n  4:   32,   454,   1066,     785,     225,     26,      1;\n  5:   64,   665,   1351,     910,     245,     27,      1;\n  6:  128,  4246,  20266,   29673,   17801,   4984,    679,    43,    1;\n  7:  256,  6305,  26335,   35574,   20181,   5418,    714,    44,    1;\n  8:  512, 38854, 354106,  928465,  987505, 505302, 136017, 20010, 1596, 64, 1;\n  9: 1024, 58025, 465751, 1132670, 1144165, 563409, 147147, 21120, 1650, 65, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (4+5*x+x^2)*P(n-2,x) + (5*x+2*x^2)*d/dx P(n-2,x) + x^2*d^2/dx^2 P(n-2,x) for n >= 2, with P(0,x) = 2 + 4*x + x^2, P(1,x) = 4 + 5*x + x^2.","mathematica":"P[0, x_] := 2 + 4*x + x^2;\nP[1, x_] := 4 + 5*x + x^2;\nP[n_, x_] := P[n, x] = Expand[(4 + 5*x + x^2)*P[n - 2, x] + (5*x + 2*x^2)*D[P[n - 2, x], x] + x^2*D[P[n - 2, x], {x, 2}]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 2 + 4*x + x^2; v[2] = 4 + 5*x + x^2; for(n = 2, N, my(i = n + 1); v[i] = (4 + 5*x + x^2)*v[i - 2] + (5*x + 2*x^2)*deriv(v[i - 2], x) + x^2*deriv(deriv(v[i - 2], x), 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-e655e253465c7aff","steps":["basis A008277"],"edges":["e-4cb5e0c11a0e9b05"],"text":"Veronese parent of A109447 \u2192 basis A008277"},"headline":"Veronese parent of A109447 \u2192 basis A008277","chain":null}