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the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"79bef7ed5bcf6089480143dc9445dceb8f068d9626fbcb3c6865b6d7f0433576","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-cedb1276760d9bb1","role":"sequence","name":"Triangular array:  the fusion of polynomial sequences P and Q given by p(n,x)=(x+2)^n and q(n,x)=(x+2)^n.","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:        3,          2;\n  2:       21,         39,          10;\n  3:      147,        573,         365,          50;\n  4:     1029,       7611,        9070,        2925,         250;\n  5:     7203,      96477,      192245,      109700,       21375,        1250;\n  6:    50421,    1193739,     3750550,     3374625,     1125625,      146875,       6250;\n  7:   352947,   14576973,    69728045,    92994500,    47265000,    10328125,     965625,     31250;\n  8:  2470629,  176688411,  1258261630,  2394519225,  1741127500,   566825000,   87368750,   6140625,   156250;\n  9: 17294403, 2132614077, 22273889765, 58970731100, 58983038625, 26844037500, 6066768750, 694750000, 38046875, 781250;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = (7*n-7+(5*n-5)*x)*P(n-1,x) + (5*n-5)*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 2*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3 + 2*x;\nP[n_, x_] := P[n, x] = Expand[Together[((7*n - 7 + (5*n - 5)*x)*P[n - 1, x] + (5*n - 5)*x*D[P[n - 1, x], x])/(n - 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = ((7*n - 7 + (5*n - 5)*x)*v[i - 1] + (5*n - 5)*x*deriv(v[i - 1], x))/(n - 1)); 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-cedb1276760d9bb1","steps":["basis A008277"],"edges":["e-f478fe63f2749dc3"],"text":"A193726 \u2192 basis A008277"},"headline":"A193726 \u2192 basis A008277","chain":null}