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the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"a34de62cb52715864a2e38849530456fd961a5b23cc695ccf28e0453f0481a40","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-1a01d480c8d600c3","role":"sequence","name":"basis_map_A013611 of 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:           5,            4;\n  2:          40,           68,           16;\n  3:         440,         1180,          576,           64;\n  4:        6160,        23000,        17392,         3968,          256;\n  5:      104720,       507640,       526800,       184640,        24320,        1024;\n  6:     2094400,     12602240,     16780960,      8015680,      1614080,      138240,       4096;\n  7:    48171200,    348638080,    570618720,    347672640,     95011840,    12400640,     745472,     16384;\n  8:  1252451200,  10651827200,  20795588800,  15494035200,   5381187840,   950476800,   86876160,   3866624,    65536;\n  9: 36321084800, 356520102400, 812044094400, 718437798400, 304129593600, 68098114560, 8406343680, 567902208, 19464192, 262144;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (3*n+2+4*x)*P(n-1,x) + 4*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 + 4*x)*P[n - 1, x] + 4*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 + 4*x)*v[i - 1] + 4*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 A013611","basis A008277"],"edges":["e-422c7a2dd7cada12","e-987c764d04ca8c0c"],"text":"basis-map coefficients of A371080, shifted \u2192 basis A013611 \u2192 basis A008277"},"headline":"basis-map coefficients of A371080, shifted \u2192 basis A013611 \u2192 basis A008277","chain":null}