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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":"d7109274539154e32d040ec3e8d4cf4adf2813801f4c1033f76633b69f33192c","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-e3a9cbbad5cb7ab4","role":"sequence","name":"Triangle read by rows: T(n,k) is the number of words over alphabet {0,1,2,3} having exactly k occurrences of the string 01, where n>=0 and k>=0.","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:      4;\n  2:     15,      4;\n  3:     56,     32;\n  4:    209,    184,      9;\n  5:    780,    928,    108;\n  6:   2911,   4364,    837,    16;\n  7:  10864,  19648,   5328,   256;\n  8:  40545,  85872,  30294,  2496,  25;\n  9: 151316, 367296, 160056, 19200, 500;\n","recurrence_sentence":"The row polynomials satisfy (1152*n-(1440*n^2+816*n)*x+(48*n^3+624*n^2+392*n)*x^2+(72*n^4+52*n^3-52*n^2-40*n)*x^3+(2*n^5+4*n^4+2*n^3)*x^4)*P(n,x) = (4032*n+576-(4536*n^2+2784*n+1704)*x-(48*n^3+108*n^2-2752*n-1660)*x^2+(240*n^4+638*n^3+252*n^2-486*n-516)*x^3+(7*n^5+75*n^4+99*n^3-55*n^2-82*n-12)*x^4+(n^5+5*n^4+7*n^3-n^2-8*n-4)*x^5)*P(n-1,x) + (864-(2592*n+2160)*x+(360*n^2+2520*n+1368)*x^2+(216*n^3+348*n^2+780*n+224)*x^3+(6*n^4-108*n^3-434*n^2-488*n-240)*x^4-(4*n^4+96*n^3+248*n^2+196*n+48)*x^5-(2*n^4+12*n^3+26*n^2+24*n+8)*x^6)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((4032*n + 576 - (4536*n^2 + 2784*n + 1704)*x - (48*n^3 + 108*n^2 - 2752*n - 1660)*x^2 + (240*n^4 + 638*n^3 + 252*n^2 - 486*n - 516)*x^3 + (7*n^5 + 75*n^4 + 99*n^3 - 55*n^2 - 82*n - 12)*x^4 + (n^5 + 5*n^4 + 7*n^3 - n^2 - 8*n - 4)*x^5)*P[n - 1, x] + (864 - (2592*n + 2160)*x + (360*n^2 + 2520*n + 1368)*x^2 + (216*n^3 + 348*n^2 + 780*n + 224)*x^3 + (6*n^4 - 108*n^3 - 434*n^2 - 488*n - 240)*x^4 - (4*n^4 + 96*n^3 + 248*n^2 + 196*n + 48)*x^5 - (2*n^4 + 12*n^3 + 26*n^2 + 24*n + 8)*x^6)*D[P[n - 1, x], x])/(1152*n - (1440*n^2 + 816*n)*x + (48*n^3 + 624*n^2 + 392*n)*x^2 + (72*n^4 + 52*n^3 - 52*n^2 - 40*n)*x^3 + (2*n^5 + 4*n^4 + 2*n^3)*x^4)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((4032*n + 576 - (4536*n^2 + 2784*n + 1704)*x - (48*n^3 + 108*n^2 - 2752*n - 1660)*x^2 + (240*n^4 + 638*n^3 + 252*n^2 - 486*n - 516)*x^3 + (7*n^5 + 75*n^4 + 99*n^3 - 55*n^2 - 82*n - 12)*x^4 + (n^5 + 5*n^4 + 7*n^3 - n^2 - 8*n - 4)*x^5)*v[i - 1] + (864 - (2592*n + 2160)*x + (360*n^2 + 2520*n + 1368)*x^2 + (216*n^3 + 348*n^2 + 780*n + 224)*x^3 + (6*n^4 - 108*n^3 - 434*n^2 - 488*n - 240)*x^4 - (4*n^4 + 96*n^3 + 248*n^2 + 196*n + 48)*x^5 - (2*n^4 + 12*n^3 + 26*n^2 + 24*n + 8)*x^6)*deriv(v[i - 1], x))/(1152*n - (1440*n^2 + 816*n)*x + (48*n^3 + 624*n^2 + 392*n)*x^2 + (72*n^4 + 52*n^3 - 52*n^2 - 40*n)*x^3 + (2*n^5 + 4*n^4 + 2*n^3)*x^4)); 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-e3a9cbbad5cb7ab4","steps":["(\u03b8+1)\u00b2"],"edges":["e-1674ad2138d71735"],"text":"A261711 \u2192 (\u03b8+1)\u00b2"},"headline":"A261711 \u2192 (\u03b8+1)\u00b2","chain":null}