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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-fc0676c2630bb222"]},{"id":166125,"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-fc0676c2630bb222"]},{"id":166126,"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":"44f2b97a248221d8a7ff600851d43375b289d3e91377937c5df191e396e23645","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-26633b980c3508e8","role":"sequence","name":"Triangle read by rows: (i,j)-th entry is binomial(i,j)*3^(i-j)*8^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:          121,             240,               64;\n  1:         1331,            5528,             3648,               512;\n  2:        14641,          115680,           142720,             47104,              4096;\n  3:       161051,         2315048,          4778880,           2790400,            552960,             32768;\n  4:      1771561,        45274320,        147550144,         135895040,          46100480,           6094848,           262144;\n  5:     19487171,       874384568,       4346048448,        5936727552,        3069480960,         679641088,         64225280,          2097152;\n  6:    214358881,     16769204160,     124338384640,      242553851904,      179481501696,       59217543168,       9226420224,        654311424,        16777216;\n  7:   2357947691,    320329750088,    3491290018560,     9484091893760,     9658135388160,     4455946715136,    1018099138560,     117650227200,      6492782592,      134217728;\n  8:  25937424601,   6105128833200,   96827468501824,   359873536430080,   491172556840960,   304518365577216,   95715422896128,   16027358330880,   1428160512000,    63082332160,   1073741824;\n  9: 285311670611, 116204947227608, 2663182680214848, 13370193523067392, 23999408235601920, 19459817099165696, 8083356875489280, 1839556391337984, 235330905047040, 16661117665280, 602369163264, 8589934592;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (11+8*x)*P(n-1,x) + 8*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 121 + 240*x + 64*x^2.","mathematica":"P[0, x_] := 121 + 240*x + 64*x^2;\nP[n_, x_] := P[n, x] = Expand[(11 + 8*x)*P[n - 1, x] + 8*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] = 121 + 240*x + 64*x^2; for(n = 1, N, my(i = n + 1); v[i] = (11 + 8*x)*v[i - 1] + 8*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-26633b980c3508e8","steps":["basis A008277"],"edges":["e-fc0676c2630bb222"],"text":"A038226 \u2192 basis A008277"},"headline":"A038226 \u2192 basis A008277","chain":null}