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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-047b920dffe4a1a4"]},{"id":166191,"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-047b920dffe4a1a4"]},{"id":166192,"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":"65b2d6c2bfb872a4ee089f99f24afdef4fe83b60a18dfa1c9184a009edeac8d1","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-29d7c5a1e19b9d74","role":"sequence","name":"Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*12^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:             484,               672,                144;\n  1:           10648,             28656,              14688,                1728;\n  2:          234256,           1102080,            1019520,              276480,               20736;\n  3:         5153632,          40281792,           60122880,            28270080,             4769280,              248832;\n  4:       113379904,        1431424512,         3249033984,          2361139200,           673090560,            77635584,             2985984;\n  5:      2494357888,       50028992256,       166632657408,        175934481408,         75450009600,         14443204608,          1212309504,           35831808;\n  6:     54875873536,     1730918031360,      8265450147840,      12203791810560,       7392714448896,       2089742893056,        287275548672,        18345885696,         429981696;\n  7:   1207269217792,    59509723548672,    400981723176960,     807005326786560,     663935513149440,     260071490617344,      52080816291840,      5391970467840,      270888468480,       5159780352;\n  8:  26559922791424,  2037817831268352,  19159275948724224,   51618089631744000,   56159549841899520,   29293088388415488,    8016454618841088,   1196518665093120,    96668484894720,    3921433067520,    61917364224;\n  9: 584318301411328, 69604525336621056, 905780507616534528, 3223760510025842688, 4550585564513894400, 3075947845952864256, 1105063794832048128, 223028433925963776, 25765105198694400, 1669808117514240, 55849462530048, 743008370688;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (22+12*x)*P(n-1,x) + 12*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 484 + 672*x + 144*x^2.","mathematica":"P[0, x_] := 484 + 672*x + 144*x^2;\nP[n_, x_] := P[n, x] = Expand[(22 + 12*x)*P[n - 1, x] + 12*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] = 484 + 672*x + 144*x^2; for(n = 1, N, my(i = n + 1); v[i] = (22 + 12*x)*v[i - 1] + 12*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-29d7c5a1e19b9d74","steps":["basis A008277"],"edges":["e-047b920dffe4a1a4"],"text":"A038314 \u2192 basis A008277"},"headline":"A038314 \u2192 basis A008277","chain":null}