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the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"6a615b3574be44a41440dca960ec550b44e87ca9bf48cae7f6598447cc4b2a94","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-4e0b19699f4cf0db","role":"sequence","name":"Triangular array T read by rows: T(n,0) = T(n,n) = 1 for all n >= 0, T(n,k) = T(n-1,k-1) + T(n-1,k) for odd n and 1< = k <= n-1, T(n,k) = T(n-1,k-1) + T(n-1,k) + T(n-2,k-1) for even n and 1 <= k <= n-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:    2,     1;\n  2:    5,     6,      1;\n  3:   10,    23,     10,      1;\n  4:   25,    96,     72,     16,      1;\n  5:   50,   313,    384,    152,     22,     1;\n  6:  125,  1206,   2161,   1280,    305,    30,     1;\n  7:  250,  3743,   9850,   8561,   3110,   515,    38,    1;\n  8:  625, 14016,  50832,  59936,  30026,  7200,   856,   48,  1;\n  9: 1250, 42673, 217344, 350512, 240092, 80426, 14048, 1288, 58, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (5+6*x+x^2)*P(n-2,x) + (6*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) = 1, P(1,x) = 2 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + x;\nP[n_, x_] := P[n, x] = Expand[(5 + 6*x + x^2)*P[n - 2, x] + (6*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] = 1; v[2] = 2 + x; for(n = 2, N, my(i = n + 1); v[i] = (5 + 6*x + x^2)*v[i - 2] + (6*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-4e0b19699f4cf0db","steps":["basis A008277"],"edges":["e-17d60577f822afda"],"text":"A026374 \u2192 basis A008277"},"headline":"A026374 \u2192 basis A008277","chain":null}