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the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"d780e26454c737d41b0ac94daf4536ef6953598606790bf70886832973cb516c","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-1a0f876fa1441737","role":"sequence","name":"basis-map coefficients of Triangle read by rows: BellMatrix(Product_{p in P(n)} p), where P(n) = {k : k mod m = 1 and 1 <= k <= m*(n + 1)} and m = 3. shifted by x ->  x + 1","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:         1,          1;\n  2:         4,          6,          1;\n  3:        28,         52,         15,         1;\n  4:       280,        600,        232,        28,        1;\n  5:      3640,       8680,       4080,       680,       45,       1;\n  6:     58240,     151200,      82120,     17000,     1580,      66,      1;\n  7:   1106560,    3082240,    1875720,    456120,    53340,    3164,     91,    1;\n  8:  24344320,   71998080,   48099520,  13278720,  1842960,  138768,   5712,  120,   1;\n  9: 608608000, 1896294400, 1370685120, 419903680, 66724560, 6006000, 315840, 9552, 153, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (3*n-2+x)*P(n-1,x) + 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[(3*n - 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; for(n = 1, N, my(i = n + 1); v[i] = (3*n - 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-1a0f876fa1441737","steps":["basis A008277"],"edges":["e-1a9b6c8547b01dec"],"text":"basis-map coefficients of A371080, shifted \u2192 basis A008277"},"headline":"basis-map coefficients of A371080, shifted \u2192 basis A008277","chain":null}