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T(n,m) = 4^m (2n+1)! / ( (2n-2m)! (2m+1)! ).","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:         1;\n  1:         7,           4;\n  2:        61,          88,           16;\n  3:       547,        1596,          720,           64;\n  4:      4921,       26352,        22240,         4864,          256;\n  5:     44287,      413620,       578272,       228480,        29440,        1024;\n  6:    398581,     6311368,     13676912,      8565504,      1954560,      165888,       4096;\n  7:   3587227,    94754412,    305630512,    282476224,     99662080,    14787584,     888832,     16384;\n  8:  32285041,  1409076960,   6588726464,   8598616576,   4356281856,   983506944,  102416384,   4587520,    65536;\n  9: 290565367, 20835210340, 138718498496, 248203981056, 172467905024, 54940121088, 8608137216, 663552000, 23003136, 262144;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(9+40*x+16*x^2)*P(n-2,x) - (40*x+32*x^2)*d/dx P(n-2,x) - 16*x^2*d^2/dx^2 P(n-2,x) + (10+8*x)*P(n-1,x) + 8*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 7 + 4*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 7 + 4*x;\nP[n_, x_] := P[n, x] = Expand[-(9 + 40*x + 16*x^2)*P[n - 2, x] - (40*x + 32*x^2)*D[P[n - 2, x], x] - 16*x^2*D[P[n - 2, x], {x, 2}] + (10 + 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] = 1; v[2] = 7 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = -(9 + 40*x + 16*x^2)*v[i - 2] - (40*x + 32*x^2)*deriv(v[i - 2], x) - 16*x^2*deriv(deriv(v[i - 2], x), x) + (10 + 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-9aec4822212db1dc","steps":["basis A008277"],"edges":["e-db4d4459bbf9061f"],"text":"A085841 \u2192 basis A008277"},"headline":"A085841 \u2192 basis A008277","chain":null}