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the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"b1bcdeb05483abfc31c265fc03d296a05e0bb37142a276cd1765966af11da232","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-5a70a6e97131d437","role":"sequence","name":"Triangle of Sulanke numbers: T(n,k) = T(n,k-1) + a(n-1,k) for n+k even and a(n,k) = a(n,k-1) + 2*a(n-1,k) for n+k odd.","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:     6,       9,        2;\n  1:    18,      57,       32,         4;\n  2:    36,     189,      185,        52,         4;\n  3:   108,    1017,     1673,       838,       148,         8;\n  4:   216,    3159,     7709,      5863,      1726,       204,        8;\n  5:   648,   16227,    60281,     68185,     30712,      6104,      528,       16;\n  6:  1296,   49329,   257351,    401206,    252457,     73440,    10328,      672,      16;\n  7:  3888,  249237,  1900115,   4125556,   3579439,   1459634,   301800,    32080,    1648,     32;\n  8:  7776,  751599,  7849697,  22527895,  25602190,  13796877,  3874034,   590520,   48560,   2000,   32;\n  9: 23328, 3773547, 56451077, 218450449, 326679880, 230563781, 85704264, 17786908, 2103680, 139120, 4736, 64;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (6+9*x+2*x^2)*P(n-2,x) + (9*x+4*x^2)*d/dx P(n-2,x) + 2*x^2*d^2/dx^2 P(n-2,x) for n >= 2, with P(0,x) = 6 + 9*x + 2*x^2, P(1,x) = 18 + 57*x + 32*x^2 + 4*x^3.","mathematica":"P[0, x_] := 6 + 9*x + 2*x^2;\nP[1, x_] := 18 + 57*x + 32*x^2 + 4*x^3;\nP[n_, x_] := P[n, x] = Expand[(6 + 9*x + 2*x^2)*P[n - 2, x] + (9*x + 4*x^2)*D[P[n - 2, x], x] + 2*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] = 6 + 9*x + 2*x^2; v[2] = 18 + 57*x + 32*x^2 + 4*x^3; for(n = 2, N, my(i = n + 1); v[i] = (6 + 9*x + 2*x^2)*v[i - 2] + (9*x + 4*x^2)*deriv(v[i - 2], x) + 2*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-5a70a6e97131d437","steps":["basis A008277"],"edges":["e-a9b5217f78edb4be"],"text":"A064861 \u2192 basis A008277"},"headline":"A064861 \u2192 basis A008277","chain":null}