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T(n, k) = RisingFactorial(n + 1, n) / (k! * (n - k)!).","uses":{"claim":"real_rooted","from_index":0,"region":{}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)"},{"position":1,"node_id":"n-dd77bf712b1df996","role":"sequence","name":"Triangle read by rows. T(n, k) = RisingFactorial(n + 1, n) / (k! * (n - k)!).","uses":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"uses_text":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1"}]}],"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:      3,       2;\n  1:      8,      14,        6;\n  2:     26,      72,       66,       20;\n  3:     90,     340,      480,      300,       70;\n  4:    322,    1540,     2940,     2800,     1330,      252;\n  5:   1176,    6804,    16380,    21000,    15120,     5796,      924;\n  6:   4356,   29568,    85932,   138600,   133980,    77616,    24948,     3432;\n  7:  16302,  126984,   432432,   840840,  1021020,   792792,   384384,   106392,   12870;\n  8:  61490,  540540,  2110680,  4804800,  7027020,  6846840,  4444440,  1853280,  450450,   48620;\n  9: 233376, 2285140, 10064340, 26254800, 44924880, 52684632, 42882840, 23921040, 8751600, 1896180, 184756;\n","recurrence_sentence":"The row polynomials satisfy (5*n^2+3*n-2+(4*n^2+2*n-2)*x)*P(n,x) = (20*n^2+2*n-6+(36*n^2+2*n-10)*x+(16*n^2-4)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 3 + 2*x.","mathematica":"P[0, x_] := 3 + 2*x;\nP[n_, x_] := P[n, x] = Expand[Together[((20*n^2 + 2*n - 6 + (36*n^2 + 2*n - 10)*x + (16*n^2 - 4)*x^2)*P[n - 1, x])/(5*n^2 + 3*n - 2 + (4*n^2 + 2*n - 2)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 3 + 2*x; for(n = 1, N, my(i = n + 1); v[i] = ((20*n^2 + 2*n - 6 + (36*n^2 + 2*n - 10)*x + (16*n^2 - 4)*x^2)*v[i - 1])/(5*n^2 + 3*n - 2 + (4*n^2 + 2*n - 2)*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-dd77bf712b1df996","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-9dcb6fc37a08fff7"],"text":"A356546 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A356546 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}