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the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"8d27c5f01c0e68756887333b848b910d0d20d1336e5ce94c93a3b59ffd7d2301","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-c6ca397ede50cf7f","role":"sequence","name":"Mirror of the triangle A193728.","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:       21,         34,          8;\n  3:      147,        458,        256,          32;\n  4:     1029,       5626,       5672,        1632,         128;\n  5:     7203,      66002,     107584,       53696,        9472,        512;\n  6:    50421,     754834,    1877768,     1450560,      432640,      51712,       2048;\n  7:   352947,    8504858,   31185856,    35071712,    15752960,    3126784,     270336,      8192;\n  8:  2470629,   94965226,  501807272,   791105952,   502604928,  147435008,   20887552,   1368064,   32768;\n  9: 17294403, 1054500002, 7906969984, 17038242176, 14724337152, 5991164928, 1237254144, 131432448, 6750208, 131072;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = (7*n-7+(4*n-4)*x)*P(n-1,x) + (4*n-4)*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[((7*n - 7 + (4*n - 4)*x)*P[n - 1, x] + (4*n - 4)*x*D[P[n - 1, x], x])/(n - 1)]];\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] = ((7*n - 7 + (4*n - 4)*x)*v[i - 1] + (4*n - 4)*x*deriv(v[i - 1], x))/(n - 1)); 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-c6ca397ede50cf7f","steps":["basis A008277"],"edges":["e-b0b41453097d536c"],"text":"A193729 \u2192 basis A008277"},"headline":"A193729 \u2192 basis A008277","chain":null}