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Stirling set numbers of order 2, T(n, n) = 1, T(n, k) = 0 if k < 0 or k > n, otherwise T(n, k) = T(n-1, k-1) + k^2*T(n-1, k), for 0 <= k <= n.","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: 1;\n   2: 1,     1;\n   3: 1,     3,       1;\n   4: 1,    11,       8,        1;\n   5: 1,    43,      63,       18,        1;\n   6: 1,   171,     526,      267,       35,       1;\n   7: 1,   683,    4565,     3916,      902,      61,      1;\n   8: 1,  2731,   40404,    58773,    21450,    2548,     98,     1;\n   9: 1, 10923,  360907,   902694,   503503,   93158,   6258,   148,   1;\n  10: 1, 43691, 3237242, 14093119, 11921845, 3242291, 336464, 13770, 213, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (x^2-x^3)*P(n-2,x) - x^3*d/dx P(n-2,x) - x^4*d^2/dx^2 P(n-2,x) + (1-x+x^2)*P(n-1,x) + 3*x*d/dx P(n-1,x) + x^2*d^2/dx^2 P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 1 + x;\nP[n_, x_] := P[n, x] = Expand[(x^2 - x^3)*P[n - 2, x] - x^3*D[P[n - 2, x], x] - x^4*D[P[n - 2, x], {x, 2}] + (1 - x + x^2)*P[n - 1, x] + 3*x*D[P[n - 1, x], x] + x^2*D[P[n - 1, x], {x, 2}]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + x; for(n = 3, N, my(i = n); v[i] = (x^2 - x^3)*v[i - 2] - x^3*deriv(v[i - 2], x) - x^4*deriv(deriv(v[i - 2], x), x) + (1 - x + x^2)*v[i - 1] + 3*x*deriv(v[i - 1], x) + x^2*deriv(deriv(v[i - 1], x), x)); 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-0164479e47952770","steps":["D+1"],"edges":["e-61f588975beb042b"],"text":"A269945 \u2192 D+1"},"headline":"A269945 \u2192 D+1","chain":null}