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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":"899ab665105da72fedc80f7162db03da8af14f4fb8f55cf2a3541522dd28889e","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-787407680a56b31c","role":"sequence","name":"Triangle read by rows: T(n,k) = k*(k+1)*binomial(n,k)/2 (1 <= k <= n).","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:  2,  12;\n  2:  3,  36,   54;\n  3:  4,  72,  216,   160;\n  4:  5, 120,  540,   800,   375;\n  5:  6, 180, 1080,  2400,  2250,    756;\n  6:  7, 252, 1890,  5600,  7875,   5292,   1372;\n  7:  8, 336, 3024, 11200, 21000,  21168,  10976,   2304;\n  8:  9, 432, 4536, 20160, 47250,  63504,  49392,  20736,  3645;\n  9: 10, 540, 6480, 33600, 94500, 158760, 164640, 103680, 36450, 5500;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+5*x+10*x^2+10*x^3+5*x^4+x^5)*P(n-5,x) - (5+20*x+30*x^2+20*x^3+5*x^4)*P(n-4,x) + (10+30*x+30*x^2+10*x^3)*P(n-3,x) - (10+20*x+10*x^2)*P(n-2,x) + (5+5*x)*P(n-1,x) for n >= 5, with P(0,x) = 1, P(1,x) = 2 + 12*x, P(2,x) = 3 + 36*x + 54*x^2, P(3,x) = 4 + 72*x + 216*x^2 + 160*x^3, P(4,x) = 5 + 120*x + 540*x^2 + 800*x^3 + 375*x^4.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + 12*x;\nP[2, x_] := 3 + 36*x + 54*x^2;\nP[3, x_] := 4 + 72*x + 216*x^2 + 160*x^3;\nP[4, x_] := 5 + 120*x + 540*x^2 + 800*x^3 + 375*x^4;\nP[n_, x_] := P[n, x] = Expand[(1 + 5*x + 10*x^2 + 10*x^3 + 5*x^4 + x^5)*P[n - 5, x] - (5 + 20*x + 30*x^2 + 20*x^3 + 5*x^4)*P[n - 4, x] + (10 + 30*x + 30*x^2 + 10*x^3)*P[n - 3, x] - (10 + 20*x + 10*x^2)*P[n - 2, x] + (5 + 5*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] = 1; v[2] = 2 + 12*x; v[3] = 3 + 36*x + 54*x^2; v[4] = 4 + 72*x + 216*x^2 + 160*x^3; v[5] = 5 + 120*x + 540*x^2 + 800*x^3 + 375*x^4; for(n = 5, N, my(i = n + 1); v[i] = (1 + 5*x + 10*x^2 + 10*x^3 + 5*x^4 + x^5)*v[i - 5] - (5 + 20*x + 30*x^2 + 20*x^3 + 5*x^4)*v[i - 4] + (10 + 30*x + 30*x^2 + 10*x^3)*v[i - 3] - (10 + 20*x + 10*x^2)*v[i - 2] + (5 + 5*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-c8c19eba9d68881f","steps":["(1/2+(1/2)n)\u03b8+1+n","(\u03b8+1)\u00b2"],"edges":["e-c94c2672fb190cd9","e-6fe488c5b3e4b6c7"],"text":"A007318 \u2192 (1/2+(1/2)n)\u03b8+1+n \u2192 (\u03b8+1)\u00b2"},"headline":"A007318 \u2192 (1/2+(1/2)n)\u03b8+1+n \u2192 (\u03b8+1)\u00b2","chain":null}