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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-f8d6f67754ace198"]},{"id":166188,"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-f8d6f67754ace198"]},{"id":166189,"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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121;\n  1:            8000,             21791,              11253,                1331;\n  2:          160000,            763521,             712327,              194326,               14641;\n  3:         3200000,          25429151,           38316465,            18134875,             3074610,              161051;\n  4:        64000000,         823503681,         1889012191,          1382629490,           396258665,            45899535,            1771561;\n  5:      1280000000,       26232614111,        88397052513,         94058497071,         40569478950,          7801310440,          657249131,          19487171;\n  6:     25600000000,      827291037441,      4001234960767,       5957467922406,       3631090120581,       1031362551450,       142337840106,        9119996028,        214358881;\n  7:    512000000000,    25927622160671,    177152069764065,     359759384455955,     297921914863650,     117294182685141,     23586042315066,     2450355855882,     123470715456,      2357947691;\n  8:  10240000000000,   809388286980801,   7725590773858111,   21015920143570330,   23024355780289105,   12074204764885725,   3318635648632227,   497130983486280,   40288751683950,   1638773645245,    25937424601;\n  9: 204800000000000, 25203676896404831, 333378083658829473, 1198825266121666711, 1704733891517776350, 1158833270949609530, 418218918196114497, 84726697533123657, 9819626000215680, 638190332307605, 21398375295825, 285311670611;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (20+11*x)*P(n-1,x) + 11*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 400 + 561*x + 121*x^2.","mathematica":"P[0, x_] := 400 + 561*x + 121*x^2;\nP[n_, x_] := P[n, x] = Expand[(20 + 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] = 400 + 561*x + 121*x^2; for(n = 1, N, my(i = n + 1); v[i] = (20 + 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-cfc5b48d32898a3e","steps":["basis A008277"],"edges":["e-f8d6f67754ace198"],"text":"A038301 \u2192 basis A008277"},"headline":"A038301 \u2192 basis A008277","chain":null}