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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":"3c93e7498a90cbd4e22e8aec05650aae841d3c496d64a400de91813826a4cad3","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-2821c21537bbbcdd","role":"sequence","name":"Triangle read by rows: T(n,k) = number of k-part order-consecutive partition of {1,2,...,n} (1 <= 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,   8;\n   3: 1,  20,   36;\n   4: 1,  36,  144,   128;\n   5: 1,  56,  369,   704,    400;\n   6: 1,  80,  765,  2336,   2800,   1152;\n   7: 1, 108, 1395,  6032,  11400,   9792,   3136;\n   8: 1, 140, 2331, 13328,  35200,  47232,  31360,   8192;\n   9: 1, 176, 3654, 26432,  91225, 169632, 174832,  94208,  20736;\n  10: 1, 216, 5454, 48384, 209025, 504072, 715792, 595968, 269568, 51200;\n","recurrence_sentence":"The row polynomials satisfy (36*n-36-(22*n^3-108*n^2+122*n-36)*x+(n^5-11*n^4+35*n^3-73*n^2+96*n-48)*x^2+(4*n^4-28*n^3+44*n^2-20*n)*x^3)*P(n,x) = (36*n-36-(22*n^3-108*n^2+122*n-324)*x+(n^5-11*n^4+35*n^3-225*n^2+760*n-384)*x^2+(10*n^4-96*n^3+238*n^2-280*n+96)*x^3+(16*n^3-80*n^2)*x^4)*P(n-1,x) + (72*x-(44*n^2-172*n+48)*x^2+(2*n^4-20*n^3+14*n^2+36*n-72)*x^3+(2*n^4-12*n^3+18*n^2-136*n+48)*x^4+(8*n^3-40*n^2)*x^5)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((36*n - 36 - (22*n^3 - 108*n^2 + 122*n - 324)*x + (n^5 - 11*n^4 + 35*n^3 - 225*n^2 + 760*n - 384)*x^2 + (10*n^4 - 96*n^3 + 238*n^2 - 280*n + 96)*x^3 + (16*n^3 - 80*n^2)*x^4)*P[n - 1, x] + (72*x - (44*n^2 - 172*n + 48)*x^2 + (2*n^4 - 20*n^3 + 14*n^2 + 36*n - 72)*x^3 + (2*n^4 - 12*n^3 + 18*n^2 - 136*n + 48)*x^4 + (8*n^3 - 40*n^2)*x^5)*D[P[n - 1, x], x])/(36*n - 36 - (22*n^3 - 108*n^2 + 122*n - 36)*x + (n^5 - 11*n^4 + 35*n^3 - 73*n^2 + 96*n - 48)*x^2 + (4*n^4 - 28*n^3 + 44*n^2 - 20*n)*x^3)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((36*n - 36 - (22*n^3 - 108*n^2 + 122*n - 324)*x + (n^5 - 11*n^4 + 35*n^3 - 225*n^2 + 760*n - 384)*x^2 + (10*n^4 - 96*n^3 + 238*n^2 - 280*n + 96)*x^3 + (16*n^3 - 80*n^2)*x^4)*v[i - 1] + (72*x - (44*n^2 - 172*n + 48)*x^2 + (2*n^4 - 20*n^3 + 14*n^2 + 36*n - 72)*x^3 + (2*n^4 - 12*n^3 + 18*n^2 - 136*n + 48)*x^4 + (8*n^3 - 40*n^2)*x^5)*deriv(v[i - 1], x))/(36*n - 36 - (22*n^3 - 108*n^2 + 122*n - 36)*x + (n^5 - 11*n^4 + 35*n^3 - 73*n^2 + 96*n - 48)*x^2 + (4*n^4 - 28*n^3 + 44*n^2 - 20*n)*x^3)); 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-1bf99ad40f056d33","steps":["basis A056242","(\u03b8+1)\u00b2"],"edges":["e-3cb5f3d64c4bb317","e-6c5bdfbf185d1b6a"],"text":"basis-map coefficients of A026729 \u2192 basis A056242 \u2192 (\u03b8+1)\u00b2"},"headline":"basis-map coefficients of A026729 \u2192 basis A056242 \u2192 (\u03b8+1)\u00b2","chain":null}