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the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"aeceaad746c0ff7ddccbcd13d7bbe23defd2877e996d1440bc420eb24b6853e0","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-66e2c3448245b0b7","role":"sequence","name":"Triangle read by rows: T(n, k) is the number of diagonal dissections of a convex n-gon into k+1 regions.","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:     11,       20,         5;\n  3:     45,      170,       105,         14;\n  4:    197,     1400,      1610,        504,         42;\n  5:    903,    11522,     22050,      12180,       2310,       132;\n  6:   4279,    95676,    287259,     249480,      80850,     10296,       429;\n  7:  20793,   803882,   3655575,    4691610,    2312310,    492492,     45045,     1430;\n  8: 103049,  6836720,  46033460,   84035952,   59135076,  18738720,   2822820,   194480,   4862;\n  9: 518859, 58815746, 577545540, 1461989672, 1412190780, 624696072, 137837700, 15461160, 831402, 16796;\n","recurrence_sentence":"The row polynomials satisfy (n+2)*P(n,x) = -(n-1)*P(n-2,x) + (6*n+3+(4*n+2)*x)*P(n-1,x) + (4*n+2)*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[(-(n - 1)*P[n - 2, x] + (6*n + 3 + (4*n + 2)*x)*P[n - 1, x] + (4*n + 2)*x*D[P[n - 1, x], x])/(n + 2)]];\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] = (-(n - 1)*v[i - 2] + (6*n + 3 + (4*n + 2)*x)*v[i - 1] + (4*n + 2)*x*deriv(v[i - 1], x))/(n + 2)); 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-66e2c3448245b0b7","steps":["basis A008277"],"edges":["e-2ea22f3ef3dd233f"],"text":"A033282 \u2192 basis A008277"},"headline":"A033282 \u2192 basis A008277","chain":null}