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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-0663a0d7f7904188"]},{"id":203917,"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-0663a0d7f7904188"]},{"id":203918,"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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binomial(i,j)*10^(i-j)*3^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,             1188,               729;\n  1:           10648,            39204,             48114,              19683;\n  2:          234256,          1149984,           2117016,            1732104,             531441;\n  3:         5153632,         31624560,          77623920,           95265720,           58458510,           14348907;\n  4:       113379904,        834888384,        2561589360,         4191691680,         3858261660,         1894055724,          387420489;\n  5:      2494357888,      21428801856,       78896952288,       161380129680,       198057431880,       145842290748,        59662755306,        10460353203;\n  6:     54875873536,     538781303808,     2314310600448,      5680580564736,      8714527002720,      8556081057216,      5250322466928,      1841022163728,       282429536481;\n  7:   1207269217792,   13334837269248,    65461928412672,    187459158636288,    345095269307712,    423526012332192,    346521282817248,    182261194209072,     55921048223238,      7625597484987;\n  8:  26559922791424,  325962688803840,  1800203031348480,   5891573557140480,  12653493207949440,  18635144542616448,  19058670554948640,  13365820908665280,   6151315304556180,   1677631446697140,   205891132094649;\n  9: 584318301411328, 7888297069052928, 48405459287370240, 178220100103499520, 437449336617680640, 751617496552196736, 922439654859514176, 808632164974249440, 496206101234198520, 202993405050353940, 49825653966905058, 5559060566555523;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (22+27*x)*P(n-1,x) for n >= 1, with P(0,x) = 484 + 1188*x + 729*x^2.","mathematica":"P[0, x_] := 484 + 1188*x + 729*x^2;\nP[n_, x_] := P[n, x] = Expand[(22 + 27*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 484 + 1188*x + 729*x^2; for(n = 1, N, my(i = n + 1); v[i] = (22 + 27*x)*v[i - 1]); 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-4397e742863fdbe2","steps":["basis A013610","basis A013610"],"edges":["e-c80be26a429cc3fc","e-0663a0d7f7904188"],"text":"A038305 \u2192 basis A013610 \u2192 basis A013610"},"headline":"A038305 \u2192 basis A013610 \u2192 basis A013610","chain":null}