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the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"57e239a2532bc216340cb4aa18385c458310ddd40674d58e357131248dfef5d3","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-8addbc07b5fd134f","role":"sequence","name":"basis_map_A007318 of Triangle read by rows: T(n,k) are the coefficients of Charlier polynomials: A046716 transposed, for 0 <= k <= n.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:       1;\n   2:       3,        1;\n   3:      11,        8,        1;\n   4:      49,       57,       15,        1;\n   5:     261,      424,      174,       24,       1;\n   6:    1631,     3425,     1930,      410,      35,      1;\n   7:   11743,    30336,    21855,     6320,     825,     48,     1;\n   8:   95901,   294553,   259161,    95235,   16835,   1491,    63,    1;\n   9:  876809,  3123632,  3251500,  1452976,  325150,  38864,  2492,   80,  1;\n  10: 8877691, 36003969, 43319844, 22831284, 6195546, 935046, 80724, 3924, 99, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n-2)*P(n-2,x) + (n+1+x)*P(n-1,x) + x*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 3 + x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 3 + x;\nP[n_, x_] := P[n, x] = Expand[-(n - 2)*P[n - 2, x] + (n + 1 + x)*P[n - 1, x] + x*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 3 + x; for(n = 3, N, my(i = n); v[i] = -(n - 2)*v[i - 2] + (n + 1 + 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, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-3272d5bc697b9cb4","steps":["x \u21a6 x+1","basis A008277"],"edges":["e-77ff853dcb30433c","e-b838be4252f4fe53"],"text":"A094816 \u2192 x \u21a6 x+1 \u2192 basis A008277"},"headline":"A094816 \u2192 x \u21a6 x+1 \u2192 basis A008277","chain":null}