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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":"3398ad4bc6b7fcdcb63d66340f110decb4c90001d0e94fde91edd5cdbf25d740","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-efe494309672b404","role":"sequence","name":"derivative of Triangle read by rows: T(n,k) = binomial(2n+1, 2k+1).","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:  10,     8;\n  3:  35,   168,      27;\n  4:  84,  1008,     972,      64;\n  5: 165,  3696,    8910,    3520,     125;\n  6: 286, 10296,   46332,   45760,    9750,     216;\n  7: 455, 24024,  173745,  320320,  170625,   22680,     343;\n  8: 680, 49504,  525096, 1555840, 1547000,  514080,   46648,   512;\n  9: 969, 93024, 1360476, 5912192, 9447750, 5860512, 1329468, 87552, 729;\n","recurrence_sentence":"The row polynomials satisfy (2*n^2-5*n+3+(4*n^4-28*n^3+67*n^2-67*n+24)*x+(8*n^6-60*n^5+186*n^4-305*n^3+279*n^2-135*n+27)*x^2)*P(n,x) = (2*n^2+n+(4*n^4-16*n^3+31*n^2-44*n+16)*x+(8*n^6-36*n^5+66*n^4-111*n^3+165*n^2-111*n+28)*x^2+(24*n^6-108*n^5+162*n^4-93*n^3+30*n^2-22*n+4)*x^3)*P(n-1,x) + ((4*n-2)*x+(8*n^3-28*n^2+20*n-6)*x^2+(16*n^5-40*n^4+20*n^3+18*n^2-14*n+6)*x^3-(16*n^5-40*n^4+28*n^3-10*n^2+10*n-2)*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n^2 + n + (4*n^4 - 16*n^3 + 31*n^2 - 44*n + 16)*x + (8*n^6 - 36*n^5 + 66*n^4 - 111*n^3 + 165*n^2 - 111*n + 28)*x^2 + (24*n^6 - 108*n^5 + 162*n^4 - 93*n^3 + 30*n^2 - 22*n + 4)*x^3)*P[n - 1, x] + ((4*n - 2)*x + (8*n^3 - 28*n^2 + 20*n - 6)*x^2 + (16*n^5 - 40*n^4 + 20*n^3 + 18*n^2 - 14*n + 6)*x^3 - (16*n^5 - 40*n^4 + 28*n^3 - 10*n^2 + 10*n - 2)*x^4)*D[P[n - 1, x], x])/(2*n^2 - 5*n + 3 + (4*n^4 - 28*n^3 + 67*n^2 - 67*n + 24)*x + (8*n^6 - 60*n^5 + 186*n^4 - 305*n^3 + 279*n^2 - 135*n + 27)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((2*n^2 + n + (4*n^4 - 16*n^3 + 31*n^2 - 44*n + 16)*x + (8*n^6 - 36*n^5 + 66*n^4 - 111*n^3 + 165*n^2 - 111*n + 28)*x^2 + (24*n^6 - 108*n^5 + 162*n^4 - 93*n^3 + 30*n^2 - 22*n + 4)*x^3)*v[i - 1] + ((4*n - 2)*x + (8*n^3 - 28*n^2 + 20*n - 6)*x^2 + (16*n^5 - 40*n^4 + 20*n^3 + 18*n^2 - 14*n + 6)*x^3 - (16*n^5 - 40*n^4 + 28*n^3 - 10*n^2 + 10*n - 2)*x^4)*deriv(v[i - 1], x))/(2*n^2 - 5*n + 3 + (4*n^4 - 28*n^3 + 67*n^2 - 67*n + 24)*x + (8*n^6 - 60*n^5 + 186*n^4 - 305*n^3 + 279*n^2 - 135*n + 27)*x^2)); 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-c30cb06ad75c5e21","steps":["D","(\u03b8+1)\u00b2"],"edges":["e-1287908e85a1078d","e-28ceca67a5a00bf8"],"text":"A103327 \u2192 D \u2192 (\u03b8+1)\u00b2"},"headline":"A103327 \u2192 D \u2192 (\u03b8+1)\u00b2","chain":null}