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T(n, k) is the coefficient of x^k in a(n+3) where a(1) = a(2) = a(3) = 1 and a(m+2) = (m*x + 2)*a(m+1) - a(m) for all m in Z.","uses":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":false,"value":"-3"},"upper":{"closed":false,"value":"-1"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-3, -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:       1;\n  1:       2,         3;\n  2:       7,        22,         17;\n  3:      33,       159,        253,        133;\n  4:     191,      1248,       3040,       3272,        1313;\n  5:    1304,     10803,      35652,      58586,       47936,       15623;\n  6:   10241,    103050,     430731,     957216,     1192795,      790190,      217409;\n  7:   90865,   1077895,    5466817,   15365947,    25849851,    26026349,    14520539,     3462921;\n  8:  898409,  12290988,   73423208,  250140448,   531547598,   721427812,   610689088,   294773904,   62115169;\n  9: 9791634, 151909423, 1045761576, 4192631612, 10787783976, 18473565894, 21053713120, 15397764356, 6557298558, 1238840459;\n","recurrence_sentence":"The row polynomials satisfy (1+x)*P(n,x) = -(1+3*x+3*x^2+x^3)*P(n-2,x) + (n+2+(3*n+4)*x+(2*n+2)*x^2)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 3*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + 3*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(1 + 3*x + 3*x^2 + x^3)*P[n - 2, x] + (n + 2 + (3*n + 4)*x + (2*n + 2)*x^2)*P[n - 1, x])/(1 + x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = (-(1 + 3*x + 3*x^2 + x^3)*v[i - 2] + (n + 2 + (3*n + 4)*x + (2*n + 2)*x^2)*v[i - 1])/(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-f3e0631eb8740f76","steps":["basis A007318","x \u21a6 x/(x+1)"],"edges":["e-3f96abc359582f71","e-7698da2a4697065d"],"text":"A358735 \u2192 basis A007318 \u2192 x \u21a6 x/(x+1)"},"headline":"A358735 \u2192 basis A007318 \u2192 x \u21a6 x/(x+1)","chain":null}