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a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"10ccb2ef23c2da6ce92975e036e2a24879692873449d93abe13e32cd9d352e64","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-57cc3445e053f3c8","role":"sequence","name":"basis_map_A013610 of Triangle read by rows: T(n, k) = binomial(n+k-1, 2*k-1).","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:       6,        12;\n   3:      35,       144,        144;\n   4:     204,      1272,       2592,       1728;\n   5:    1189,      9936,      30672,      41472,       20736;\n   6:    6930,     72612,     300672,     615168,      622080,      248832;\n   7:   40391,    508896,    2644704,    7257600,    11093760,     8957952,     2985984;\n   8:  235416,   3465456,   21674304,   74666880,   153031680,   186624000,   125411328,    35831808;\n   9: 1372105,  23108832,  168986592,  700835328,  1803098880,  2947166208,  2988969984,  1719926784,   429981696;\n  10: 7997214, 151652796, 1269551232, 6158184192, 19075585536, 39133559808, 53174403072, 46151368704, 23219011584, 5159780352;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -P(n-2,x) + (6+12*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 6 + 12*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 6 + 12*x;\nP[n_, x_] := P[n, x] = Expand[-P[n - 2, x] + (6 + 12*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 6 + 12*x; for(n = 3, N, my(i = n); v[i] = -v[i - 2] + (6 + 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, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-976cdb6e4a693ec8","steps":["basis A013610","basis A013611"],"edges":["e-1c0d9915253416a6","e-b3947f18ec282d89"],"text":"A078812 \u2192 basis A013610 \u2192 basis A013611"},"headline":"A078812 \u2192 basis A013610 \u2192 basis A013611","chain":null}