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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-8943adc32758c401"]},{"id":203993,"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-8943adc32758c401"]},{"id":203994,"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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binomial(i,j)*11^(i-j)*5^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:               961,               2790,                2025;\n  1:             29791,             129735,              188325,                91125;\n  2:            923521,            5362380,            11676150,             11299500,              4100625;\n  3:          28629151,          207792225,           603267750,            875711250,            635596875,            184528125;\n  4:         887503681,         7729870770,         28051950375,          54294097500,          59110509375,          34322231250,            8303765625;\n  5:       27512614111,       279563659515,       1217454646275,        2945454789375,        4275660178125,        3723962090625,         1801917140625,          373669453125;\n  6:      852891037441,      9904541079960,      50321458712700,      146094557553000,      265090931043750,      307847532825000,       223437725437500,        92670024375000,       16815125390625;\n  7:    26439622160671,    345420870163605,    2005669568691900,     6793396926214500,    14792073952241250,    21472365414543750,     20779708465687500,     12927468400312500,     4691419983984375,      756680642578125;\n  8:   819628286980801,  11897829972301950,   77719695786811125,   300850435303785000,   764257154199131250,  1331286655701712500,   1610427406090781250,   1335838401365625000,   727170097517578125,   234570999199218750,    34050628916015625;\n  9: 25408476896404831, 405716002055496495, 2944712918144732625, 12823749804823835625, 37230241368843393750, 75661458265713993750, 109831149095391281250, 113880223716419531250, 82655001084498046875, 39994355363466796875, 11611264460361328125, 1532278301220703125;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (31+45*x)*P(n-1,x) for n >= 1, with P(0,x) = 961 + 2790*x + 2025*x^2.","mathematica":"P[0, x_] := 961 + 2790*x + 2025*x^2;\nP[n_, x_] := P[n, x] = Expand[(31 + 45*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] = 961 + 2790*x + 2025*x^2; for(n = 1, N, my(i = n + 1); v[i] = (31 + 45*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-2bc1923905556bf6","steps":["basis A013610","basis A013610"],"edges":["e-fd6222a072627022","e-8943adc32758c401"],"text":"A038319 \u2192 basis A013610 \u2192 basis A013610"},"headline":"A038319 \u2192 basis A013610 \u2192 basis A013610","chain":null}