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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-e469bcf1fd1b08af"]},{"id":166146,"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-e469bcf1fd1b08af"]},{"id":166147,"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":"1aa1b2891b7b3efe1d6e4594fceb9f90768a384ec8ff29e480fd47b4ab6fccc2","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-b1e393c17b0898c2","role":"sequence","name":"Triangle of coefficients in expansion of (1+11x)^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:          144,             385,               121;\n  1:         1728,           10439,              8349,               1331;\n  2:        20736,          259105,            398695,             151734,              14641;\n  3:       248832,         6187511,          16405785,           11213675,            2488970,             161051;\n  4:      2985984,       145049905,         625859311,          685079010,          262732745,           38169087,            1771561;\n  5:     35831808,      3368993639,       22874765529,        37713007871,        22248902830,         5447389024,          558041715,          19487171;\n  6:    429981696,     77881003585,      814800958015,      1948707775014,      1660781645061,       609712995738,       103448533034,        7872817084,        214358881;\n  7:   5159780352,   1795992881111,    28559923611945,     96654660413795,    114439557648570,     59119368810117,     14775828529770,     1838614583850,     108036876024,      2357947691;\n  8:  61917364224,  41364593849425,   990793324498351,   4663618878352170,   7471816492871665,   5219832844412109,   1802827682233347,   326170811790120,   31028448024750,   1450137829965,    25937424601;\n  9: 743008370688, 952066749543239, 34141983565287609, 220761576095329511, 469721531262687110, 431918781997199618, 198038720502734265, 48860306753887497, 6690723732166320, 502278227398365, 19115881930937, 285311670611;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (12+11*x)*P(n-1,x) + 11*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 144 + 385*x + 121*x^2.","mathematica":"P[0, x_] := 144 + 385*x + 121*x^2;\nP[n_, x_] := P[n, x] = Expand[(12 + 11*x)*P[n - 1, x] + 11*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] = 144 + 385*x + 121*x^2; for(n = 1, N, my(i = n + 1); v[i] = (12 + 11*x)*v[i - 1] + 11*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-b1e393c17b0898c2","steps":["basis A008277"],"edges":["e-e469bcf1fd1b08af"],"text":"A013618 \u2192 basis A008277"},"headline":"A013618 \u2192 basis A008277","chain":null}