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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-cfdeae139dd131be"]},{"id":166023,"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-cfdeae139dd131be"]},{"id":166024,"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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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,           1;\n  1:         8,          22,            10,              1;\n  2:        64,         386,           446,            167,              23,               1;\n  3:       512,        6238,         14242,          11137,            3721,             575,              40,               1;\n  4:      4096,       97154,        398462,         552695,          337631,          103177,           16696,            1434,              61,              1;\n  5:     32768,     1485982,      10441570,       23824561,        23789785,        12117311,         3427320,          562367,           54130,           2980,             86,             1;\n  6:    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1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (8+7*x+x^2)*P(n-1,x) + (7*x+2*x^2)*d/dx P(n-1,x) + x^2*d^2/dx^2 P(n-1,x) for n >= 1, with P(0,x) = 1 + x.","mathematica":"P[0, x_] := 1 + x;\nP[n_, x_] := P[n, x] = Expand[(8 + 7*x + x^2)*P[n - 1, x] + (7*x + 2*x^2)*D[P[n - 1, x], x] + x^2*D[P[n - 1, x], {x, 2}]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1 + x; for(n = 1, N, my(i = n + 1); v[i] = (8 + 7*x + x^2)*v[i - 1] + (7*x + 2*x^2)*deriv(v[i - 1], x) + x^2*deriv(deriv(v[i - 1], x), 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-7e353d0275e3ebc5","steps":["basis A008277"],"edges":["e-cfdeae139dd131be"],"text":"A299989 \u2192 basis A008277"},"headline":"A299989 \u2192 basis A008277","chain":null}