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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-4c5ee4490c6f16fe"]},{"id":165978,"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-4c5ee4490c6f16fe"]},{"id":165979,"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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    6,           9,             2;\n  2:       36,         189,           185,              52,               4;\n  3:      216,        3159,          7709,            5863,            1726,              204,                8;\n  4:     1296,       49329,        257351,          401206,          252457,            73440,            10328,              672,               16;\n  5:     7776,      751599,       7849697,        22527895,        25602190,         13796877,          3874034,           590520,            48560,            2000,              32;\n  6:    46656,    11343969,     229577855,      1148703322,      2178529729,       1940626347,        916197893,        245786436,         38823708,         3634000,          195952,           5568,            64;\n  7:   279936,   170579439,    6575744849,     55617160963,    168364887586,     233357647446,     170578970681,      71721107871,      18302998438,      2922798900,       295260168,       18679248,        711200,        14784,         128;\n  8:  1679616,  2561211009,  186338948351,   2614901064646,  12293194450597,   25555902429162,   27573578032826,   17069150831628,    6472408635809,   1569031697600,    249623753744,    26384101632,    1849354080,     84327936,     2387712,     37888,     256;\n  9: 10077696, 38433281679, 5250789656177, 120843432453055, 866636433993670, 2638146784447959, 4074536923286012, 3572619957180666, 1916463377639598, 661953377877025, 152447399696962, 23944460359056, 2596660736224, 194864037984, 10027017408, 345112320, 7550976, 94464, 512;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (6+9*x+2*x^2)*P(n-1,x) + (9*x+4*x^2)*d/dx P(n-1,x) + 2*x^2*d^2/dx^2 P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(6 + 9*x + 2*x^2)*P[n - 1, x] + (9*x + 4*x^2)*D[P[n - 1, x], x] + 2*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; for(n = 1, N, my(i = n + 1); v[i] = (6 + 9*x + 2*x^2)*v[i - 1] + (9*x + 4*x^2)*deriv(v[i - 1], x) + 2*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-eed4f744ce063213","steps":["basis A008277"],"edges":["e-4c5ee4490c6f16fe"],"text":"A200536 \u2192 basis A008277"},"headline":"A200536 \u2192 basis A008277","chain":null}