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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-86a79a465fd2222b"]},{"id":166176,"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-86a79a465fd2222b"]},{"id":166177,"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":"657854ae03619ad0387c55bfbb9e4d06070d13db0b2d6b432ced7d5030447c3c","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-7dcf11eddf2b9a8a","role":"sequence","name":"Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*7^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:            324,              301,                49;\n  1:           5832,             9793,              3675,               343;\n  2:         104976,           285649,            186151,             39102,              2401;\n  3:        1889568,          7876057,           7956375,           2828035,            384160,             16807;\n  4:       34012224,        210128401,         309736399,         165987990,          37467605,           3579891,           117649;\n  5:      612220032,       5491295593,       11382463575,        8641686403,        2885425760,         452007458,         32118177,          823543;\n  6:    11019960576,     141567930049,      402677903551,      416703014742,      193221389781,       44154375594,       5091142826,       280004620,        5764801;\n  7:   198359290368,    3616337975257,    13876668423975,    19070162899795,    11805105033120,     3692731634949,     614549198718,     54398309322,     2386627614,      40353607;\n  8:  3570467226624,   91796964414001,   469367755393999,   840873032059830,   676525971822085,   278350511884137,   62722073367723,   7946531115600,   557398608690,   19975035465,    282475249;\n  9: 64268410079232, 2319917380936393, 15662346923505975, 36079622538091363, 37006305928234720, 19488258932613856, 5711777985252339, 971472098319261, 96873214852260, 5519768133495, 164683070167, 1977326743;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (18+7*x)*P(n-1,x) + 7*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 324 + 301*x + 49*x^2.","mathematica":"P[0, x_] := 324 + 301*x + 49*x^2;\nP[n_, x_] := P[n, x] = Expand[(18 + 7*x)*P[n - 1, x] + 7*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] = 324 + 301*x + 49*x^2; for(n = 1, N, my(i = n + 1); v[i] = (18 + 7*x)*v[i - 1] + 7*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-7dcf11eddf2b9a8a","steps":["basis A008277"],"edges":["e-86a79a465fd2222b"],"text":"A038321 \u2192 basis A008277"},"headline":"A038321 \u2192 basis A008277","chain":null}