{"id":"n-25622692aaaa49c0","revision":0,"first_index":0,"fingerprint":"f6-a8a66b2e4166f02a","name":"basis_map_A013611 of basis_map_A013609 of Coefficients of raising factorial polynomials, T(n,k) = [x^k] p(x, n) where p(x, n) = (m*x + n - 1)*p(x, n - 1) with p[x, 0] = 1, p[x, -1] = 0, p[x, 1] = m*x and m = 2. 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a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"43c7f1bd6a88d0de755fa8827ef42b3be7e872c15c58dc27f1308f856239eeee","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-52d9fa43425cb9e1","role":"sequence","name":"basis_map_A013609 of Coefficients of raising factorial polynomials, T(n,k) = [x^k] p(x, n) where p(x, n) = (m*x + n - 1)*p(x, n - 1) with p[x, 0] = 1, p[x, -1] = 0, p[x, 1] = m*x and m = 2. Triangle read by rows, for n >= 0 and 0 <= k <= n.","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:         6,          16;\n  2:        42,         208,         256;\n  3:       336,        2336,        5376,         4096;\n  4:      3024,       26400,       85760,       122880,        65536;\n  5:     30240,      312384,     1280000,      2600960,      2621440,       1048576;\n  6:    332640,     3920064,    19078144,     49090560,     70451200,      53477376,      16777216;\n  7:   3991680,    52363008,   291658752,    894337024,   1630863360,    1768947712,    1056964608,    268435456;\n  8:  51891840,   744585984,  4629371904,  16292921344,  35510616064,   49090134016,   42043703296,  20401094656,   4294967296;\n  9: 726485760, 11254473216, 76724582400, 302170849280, 757835366400, 1255431733248, 1374053990400, 958314577920, 386547056640, 68719476736;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (n+5+16*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[(n + 5 + 16*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] = (n + 5 + 16*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-60c7e36cbe887b7e","steps":["basis A013609","basis A013611"],"edges":["e-7070cc8d62757598","e-707bd7142375a18b"],"text":"A137320 \u2192 basis A013609 \u2192 basis A013611"},"headline":"A137320 \u2192 basis A013609 \u2192 basis A013611","chain":null}