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a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"7248fb26ddbabfc0df4d46c6ff0fc33e9e034160a65a682d32c4043b81617210","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-9e472291b944d56e","role":"sequence","name":"Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*12^j.","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:            23,             48;\n  2:           529,           2208,            2304;\n  3:         12167,          76176,          158976,           110592;\n  4:        279841,        2336064,         7312896,         10174464,          5308416;\n  5:       6436343,       67161840,       280327680,        585031680,        610467840,        254803968;\n  6:     148035889,     1853666784,      9671304960,      26911457280,      42122280960,      35162947584,       12230590464;\n  7:    3404825447,    49740058704,    311416019712,    1083186155520,    2260562411520,    2830617280512,     1969125064704,      587068342272;\n  8:   78310985281,  1307452971648,   9550091271168,   39861250523136,  103985870929920,  173611193204736,   181159505952768,   108020574978048,   28179280429056;\n  9: 1801152661463, 33830345641392, 282409841875968, 1375213143048192, 4305015056498688, 8984379248345088, 12500005910740992, 11180129510227968, 5833111048814592, 1352605460594688;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (23+48*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(23 + 48*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] = 1; for(n = 1, N, my(i = n + 1); v[i] = (23 + 48*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-9e472291b944d56e","steps":["basis A013611"],"edges":["e-c0f65529a8792e3f"],"text":"A038326 \u2192 basis A013611"},"headline":"A038326 \u2192 basis A013611","chain":null}