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the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"18fcdbc54d8ba4aec0c544c0bccdf58c6aecc46721c90cff2004338944e9e03e","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-ecd77b030b8fab94","role":"sequence","name":"Triangle of coefficients of (x+1)*(x+3)*(x+6)*...*(x+n(n+1)/2).","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:           8,            13,              3;\n  3:          56,           217,            135,              18;\n  4:         616,          5117,           6355,            2088,             180;\n  5:        9856,        167867,         369085,          222693,           45000,           2700;\n  6:      216832,       7425257,       27146647,        26679690,         9446553,        1287900,          56700;\n  7:     6288128,     429310945,     2515372191,      3774911086,      2078995293,      482158584,       47231100,       1587600;\n  8:   232660736,   31566071593,   289630762839,    637915506346,    512194947129,   179472243276,    29307177324,    2159136000,     57153600;\n  9: 10702393856, 2882982248083, 40810256967789, 128496090976381, 144462255836724, 71685750548601, 17337318981804, 2098271075580, 120365481600, 2571912000;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = ((1/2)*n^2+(1/2)*n+1+((1/2)*n^2+(1/2)*n)*x)*P(n-1,x) + ((1/2)*n^2+(1/2)*n)*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[((1/2)*n^2 + (1/2)*n + 1 + ((1/2)*n^2 + (1/2)*n)*x)*P[n - 1, x] + ((1/2)*n^2 + (1/2)*n)*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; for(n = 1, N, my(i = n + 1); v[i] = ((1/2)*n^2 + (1/2)*n + 1 + ((1/2)*n^2 + (1/2)*n)*x)*v[i - 1] + ((1/2)*n^2 + (1/2)*n)*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-ecd77b030b8fab94","steps":["basis A008277"],"edges":["e-cebfd0c72bfddbbf"],"text":"A128813 \u2192 basis A008277"},"headline":"A128813 \u2192 basis A008277","chain":null}