{"id":"n-b7c4a9cec98c1a71","revision":0,"first_index":0,"fingerprint":"f6-638cb76645443454","name":"theta_plus_one_squared of derivative of Triangle T(n,k), 0 <= k <= n, read by rows, giving coefficients of the polynomial (x+1)(x+2)...(x+n), expanded in increasing powers of x. 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Craven and G. Csordas, Multiplier sequences for fields, Illinois J. Math. 21 (1977); Polya--Schur, Theorem 40 of P. Branden, arXiv:1410.6601: sum_k f(k) x^k / k! = f(theta) e^x = C e^x x^m prod (1 + x/a_j), a_j > 0); f(k) has one sign for k >= 0, so coefficients of one sign stay of one sign; applied to f(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.isMultiplierSequence_eval_of_roots_nonpos","RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos","RealRooted.isMultiplierSequence_iff_preserves"],"certificate_hash":"5c45c9e6e11210967af72bfe26f6a7376c77b904cb73bc4365c7ef44f54f0d76","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-a34e8f08502aec80","role":"sequence","name":"derivative of Triangle T(n,k), 0 <= k <= n, read by rows, giving coefficients of the polynomial (x+1)(x+2)...(x+n), expanded in increasing powers of x. T(n,k) is also the unsigned Stirling number |s(n+1, k+1)|, denoting the number of permutations on n+1 elements that contain exactly k+1 cycles.","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:       0;\n  1:       1;\n  2:       3,       8;\n  3:      11,      48,       27;\n  4:      50,     280,      270,       64;\n  5:     274,    1800,     2295,      960,     125;\n  6:    1764,   12992,    19845,    11200,    2625,     216;\n  7:   13068,  105056,   182763,   125440,   40250,    6048,    343;\n  8:  109584,  944992,  1816668,  1436736,  567000,  117936,  12348,   512;\n  9: 1026576, 9381600, 19539360, 17236800, 7909125, 2041200, 298410, 23040, 729;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n^4-12*n^3+54*n^2-108*n+81+(4*n^3-36*n^2+108*n-108)*x+(6*n^2-36*n+54)*x^2+(4*n-12)*x^3+x^4)*P(n-4,x) + (4*n^3-30*n^2+76*n-65+(12*n^2-60*n+76)*x+(12*n-30)*x^2+4*x^3)*P(n-3,x) - (6*n^2-24*n+25+(12*n-24)*x+6*x^2)*P(n-2,x) + (4*n-6+4*x)*P(n-1,x) for n >= 4, with P(0,x) = 0, P(1,x) = 1, P(2,x) = 3 + 8*x, P(3,x) = 11 + 48*x + 27*x^2.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 1;\nP[2, x_] := 3 + 8*x;\nP[3, x_] := 11 + 48*x + 27*x^2;\nP[n_, x_] := P[n, x] = Expand[-(n^4 - 12*n^3 + 54*n^2 - 108*n + 81 + (4*n^3 - 36*n^2 + 108*n - 108)*x + (6*n^2 - 36*n + 54)*x^2 + (4*n - 12)*x^3 + x^4)*P[n - 4, x] + (4*n^3 - 30*n^2 + 76*n - 65 + (12*n^2 - 60*n + 76)*x + (12*n - 30)*x^2 + 4*x^3)*P[n - 3, x] - (6*n^2 - 24*n + 25 + (12*n - 24)*x + 6*x^2)*P[n - 2, x] + (4*n - 6 + 4*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; v[3] = 3 + 8*x; v[4] = 11 + 48*x + 27*x^2; for(n = 4, N, my(i = n + 1); v[i] = -(n^4 - 12*n^3 + 54*n^2 - 108*n + 81 + (4*n^3 - 36*n^2 + 108*n - 108)*x + (6*n^2 - 36*n + 54)*x^2 + (4*n - 12)*x^3 + x^4)*v[i - 4] + (4*n^3 - 30*n^2 + 76*n - 65 + (12*n^2 - 60*n + 76)*x + (12*n - 30)*x^2 + 4*x^3)*v[i - 3] - (6*n^2 - 24*n + 25 + (12*n - 24)*x + 6*x^2)*v[i - 2] + (4*n - 6 + 4*x)*v[i - 1]); 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-efe23a76953cf69d","steps":["D","(\u03b8+1)\u00b2"],"edges":["e-04f1d27fb1fcfee8","e-316df64820a80e2f"],"text":"A130534 \u2192 D \u2192 (\u03b8+1)\u00b2"},"headline":"A130534 \u2192 D \u2192 (\u03b8+1)\u00b2","chain":null}