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0]","lean_route":null,"lean_declarations":[],"certificate_hash":"11dcf6d40eec87c72bcf9772b7c9a17e0a71f85be3ea92bb81f528403465bc03","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"previous_interlaces_next"}],"premises":[]}],"used_by":[],"used_by_total":0,"links":[],"oeis":[{"oeis_id":"A051151","relation":"tail"}],"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:                 72,              -18,                 1;\n  1:              -1296,              396,               -36,               1;\n  2:              31104,           -10800,              1260,             -60,               1;\n  3:            -933120,           355104,            -48600,            3060,             -90,            1;\n  4:           33592320,        -13716864,           2104704,         -158760,            6300,         -126,            1;\n  5:        -1410877440,        609700608,        -102114432,         8772624,         -423360,        11592,         -168,         1;\n  6:        67722117120,     -30676506624,        5511193344,      -523200384,        29093904,      -979776,        19656,      -216,        1;\n  7:     -3656994324480,    1724253474816,     -328280947200,     33764014080,     -2094271200,     82001808,     -2041200,     31320,     -270,     1;\n  8:    219419659468800, -107112202813440,    21421110306816,  -2354121792000,    159420286080,  -7014379680,    204473808,  -3920400,    47520,  -330,    1;\n  9: -14481697524940800, 7288825045155840, -1520905483063296, 176793148578816, -12875860673280, 622369344960, -20509651008, 463220208, -7056720, 69300, -396, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(6*n+12-x)*P(n-1,x) for n >= 1, with P(0,x) = 72 - 18*x + x^2.","mathematica":"P[0, x_] := 72 - 18*x + x^2;\nP[n_, x_] := P[n, x] = Expand[-(6*n + 12 - x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 72 - 18*x + x^2; for(n = 1, N, my(i = n + 1); v[i] = -(6*n + 12 - x)*v[i - 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-380b79aacd7d0dd2","steps":[],"edges":[],"text":"A051151"},"headline":"Triangle read by rows: T(n, m) = S1(n, m)*6^(n-m), where S1 are the signed Stirling numbers of first kind A008275 (n >= 1, 1 <= m <= n).","chain":null}