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0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-72f5203b9bd4f955"]},{"id":166164,"position":1,"claim":"one_signed_coefficients","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":0},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 0: the coefficients of P_n have one sign","region_exact":null,"edges":["e-72f5203b9bd4f955"]},{"id":166165,"position":2,"claim":"hadamard_kernel","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":0},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 0, are Hadamard kernels (preserve 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2)","lean_route":null,"lean_declarations":[],"certificate_hash":"e78f65ef5a04ffac5b58b6dac39eba1a3ee881ae1c47e167f15dfaed8a1a079c","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-1f2dec04776afb2b","role":"sequence","name":"Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*6^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:            289,             240,               36;\n  1:           4913,            7254,             2484,               216;\n  2:          83521,          196320,           115560,             22464,             1296;\n  3:        1419857,         5016486,          4529160,           1479600,           187920,             7776;\n  4:       24137569,       123898320,        161444556,          78960960,         16582320,          1492992,            46656;\n  5:      410338673,      2994486774,       5425282044,        3732300936,       1153640880,        169664544,         11430720,          279936;\n  6:     6975757441,     71335227840,     175300099920,      163182225024,      69693081696,      14896078848,       1623815424,        85100544,        1679616;\n  7:   118587876497,   1682564784966,    5511714264720,     6763178475360,    3836509699680,    1118274196032,     175438690560,     14763824640,      619778304,      10077696;\n  8:  2015993900449,  39410517313200,  169935102386676,   269781532225920,  197875968539040,   75577945411584,   16007895775872,   1923697797120,   128868537600,    4434186240,    60466176;\n  9: 34271896307633, 918537861606294, 5164581073092804, 10461964242227256, 9731603903456160, 4739419245578688, 1301886148590720, 209545544685312, 19918641726720, 1088038448640, 31261012992, 362797056;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (17+6*x)*P(n-1,x) + 6*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 289 + 240*x + 36*x^2.","mathematica":"P[0, x_] := 289 + 240*x + 36*x^2;\nP[n_, x_] := P[n, x] = Expand[(17 + 6*x)*P[n - 1, x] + 6*x*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 289 + 240*x + 36*x^2; for(n = 1, N, my(i = n + 1); v[i] = (17 + 6*x)*v[i - 1] + 6*x*deriv(v[i - 1], x)); 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-1f2dec04776afb2b","steps":["basis A008277"],"edges":["e-72f5203b9bd4f955"],"text":"A038320 \u2192 basis A008277"},"headline":"A038320 \u2192 basis A008277","chain":null}