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and k rises.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:    7,       2;\n   2:   21,      42,         3;\n   3:   51,     322,       153,          4;\n   4:  113,    1626,      2439,        452,          5;\n   5:  239,    6722,     22893,      13444,       1195,          6;\n   6:  493,   24842,    161499,     215332,      62105,       2958,          7;\n   7: 1003,   85730,    960321,    2428036,    1600535,     257190,       7021,         8;\n   8: 2025,  283098,   5114079,   21976068,   27470085,   10228158,     990843,     16200,        9;\n   9: 4071,  907810,  25268037,  171696452,  363026515,  257544678,   58958753,   3631240,    36639,    10;\n  10: 8165, 2853450, 118423347, 1209833252, 4010144065, 4812172878, 2117208191, 315795592, 12840525, 81650, 11;\n","recurrence_sentence":"The row polynomials satisfy (2*n+1)*P(n,x) = -(2*n^3+5*n^2+3*n)*x^2*P(n-4,x) - ((4*n^2+10*n+6)*x+(4*n^3+6*n^2-3*n-5)*x^2)*P(n-3,x) - ((2*n^2+5*n+3)*x^2-(2*n^2+5*n+3)*x^3)*d/dx P(n-3,x) - (4*n+6-(8*n+10)*x+(2*n^3+n^2-2*n-1)*x^2)*P(n-2,x) - ((2*n+3)*x+(2*n^2+n-2)*x^2-(2*n^2+3*n+1)*x^3)*d/dx P(n-2,x) + (6*n+5+(4*n^2+2*n)*x)*P(n-1,x) + ((2*n+1)*x-(2*n+1)*x^2)*d/dx P(n-1,x) for n >= 5, with P(1,x) = 7 + 2*x, P(2,x) = 21 + 42*x + 3*x^2, P(3,x) = 51 + 322*x + 153*x^2 + 4*x^3, P(4,x) = 113 + 1626*x + 2439*x^2 + 452*x^3 + 5*x^4.","mathematica":"P[1, x_] := 7 + 2*x;\nP[2, x_] := 21 + 42*x + 3*x^2;\nP[3, x_] := 51 + 322*x + 153*x^2 + 4*x^3;\nP[4, x_] := 113 + 1626*x + 2439*x^2 + 452*x^3 + 5*x^4;\nP[n_, x_] := P[n, x] = Expand[Together[(-(2*n^3 + 5*n^2 + 3*n)*x^2*P[n - 4, x] - ((4*n^2 + 10*n + 6)*x + (4*n^3 + 6*n^2 - 3*n - 5)*x^2)*P[n - 3, x] - ((2*n^2 + 5*n + 3)*x^2 - (2*n^2 + 5*n + 3)*x^3)*D[P[n - 3, x], x] - (4*n + 6 - (8*n + 10)*x + (2*n^3 + n^2 - 2*n - 1)*x^2)*P[n - 2, x] - ((2*n + 3)*x + (2*n^2 + n - 2)*x^2 - (2*n^2 + 3*n + 1)*x^3)*D[P[n - 2, x], x] + (6*n + 5 + (4*n^2 + 2*n)*x)*P[n - 1, x] + ((2*n + 1)*x - (2*n + 1)*x^2)*D[P[n - 1, x], x])/(2*n + 1)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 7 + 2*x; v[2] = 21 + 42*x + 3*x^2; v[3] = 51 + 322*x + 153*x^2 + 4*x^3; v[4] = 113 + 1626*x + 2439*x^2 + 452*x^3 + 5*x^4; for(n = 5, N, my(i = n); v[i] = (-(2*n^3 + 5*n^2 + 3*n)*x^2*v[i - 4] - ((4*n^2 + 10*n + 6)*x + (4*n^3 + 6*n^2 - 3*n - 5)*x^2)*v[i - 3] - ((2*n^2 + 5*n + 3)*x^2 - (2*n^2 + 5*n + 3)*x^3)*deriv(v[i - 3], x) - (4*n + 6 - (8*n + 10)*x + (2*n^3 + n^2 - 2*n - 1)*x^2)*v[i - 2] - ((2*n + 3)*x + (2*n^2 + n - 2)*x^2 - (2*n^2 + 3*n + 1)*x^3)*deriv(v[i - 2], x) + (6*n + 5 + (4*n^2 + 2*n)*x)*v[i - 1] + ((2*n + 1)*x - (2*n + 1)*x^2)*deriv(v[i - 1], x))/(2*n + 1)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-09153823987c7e65","steps":["D"],"edges":["e-7b3ba07b708f36d0"],"text":"A046739 \u2192 D"},"headline":"A046739 \u2192 D","chain":null}