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the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"cefd407b4cb8e2ff11d528c3b7903cfa806a824d18f5ab767d26dfea60a9cd89","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-8c29b5d8d22f0df0","role":"sequence","name":"Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + ((n+1)*(n+2)/2)*T(n-2, k-1), 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:       1;\n  1:       2,         1;\n  2:      10,        11,         1;\n  3:      40,        83,        25,         1;\n  4:     230,       769,       393,        45,        1;\n  5:    1300,      6863,      5659,      1227,       72,        1;\n  6:    9040,     71393,     84043,     27838,     3059,      107,      1;\n  7:   64880,    764155,   1265805,    603645,   103324,     6616,    151,     1;\n  8:  536560,   9189515,  20385865,  13076805,  3164574,   316181,  12954,   205,   1;\n  9: 4641520, 115730555, 341619325, 288191065, 93678824, 13243941, 841828, 23544, 270, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = ((1/2)*n^2+(3/2)*n+1+((1/2)*n^2+(3/2)*n+1)*x)*P(n-2,x) + ((1/2)*n^2+(3/2)*n+1)*x*d/dx P(n-2,x) + (2+x)*P(n-1,x) + x*d/dx P(n-1,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[((1/2)*n^2 + (3/2)*n + 1 + ((1/2)*n^2 + (3/2)*n + 1)*x)*P[n - 2, x] + ((1/2)*n^2 + (3/2)*n + 1)*x*D[P[n - 2, x], x] + (2 + x)*P[n - 1, x] + x*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] = 1; v[2] = 2 + x; for(n = 2, N, my(i = n + 1); v[i] = ((1/2)*n^2 + (3/2)*n + 1 + ((1/2)*n^2 + (3/2)*n + 1)*x)*v[i - 2] + ((1/2)*n^2 + (3/2)*n + 1)*x*deriv(v[i - 2], x) + (2 + x)*v[i - 1] + 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 - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-8c29b5d8d22f0df0","steps":["basis A008277"],"edges":["e-d61a33ddedddac02"],"text":"A154227 \u2192 basis A008277"},"headline":"A154227 \u2192 basis A008277","chain":null}