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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":"7f4aa75acebb65e20e5c8dc1603b1cb77d085c11ed84cd9efa2f806b42dff96e","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-299efe8218728fe3","role":"sequence","name":"Triangle read by rows: T(n,k) is the number of induced subgraphs of the Lucas cube Lambda(n) that are isomorphic to the hypercube Q(k).","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:  1;\n  2:  3,   8;\n  3:  4,  12;\n  4:  7,  32,   18;\n  5: 11,  60,   45;\n  6: 18, 120,  135,   32;\n  7: 29, 224,  315,  112;\n  8: 47, 416,  720,  384,  50;\n  9: 76, 756, 1539, 1056, 225;\n","recurrence_sentence":"The row polynomials satisfy (100*n-100+(125*n^2-65*n-60)*x+(75*n^3-45*n^2-34*n+4)*x^2+(15*n^4+7*n^3-29*n^2-5*n+12)*x^3+(n^5+n^4-n^3-n^2)*x^4)*P(n,x) = (300*n-300+(550*n^2+80*n-630)*x+(300*n^3+640*n^2-472*n-468)*x^2+(55*n^4+351*n^3+173*n^2-407*n-172)*x^3+(3*n^5+53*n^4+113*n^3+3*n^2-124*n-48)*x^4+(2*n^5+10*n^4+14*n^3-2*n^2-16*n-8)*x^5)*P(n-1,x) - (125+(375*n+450)*x+(300*n^2+1200*n+685)*x^2+(75*n^3+800*n^2+1530*n+594)*x^3+(5*n^4+165*n^3+773*n^2+993*n+326)*x^4+(9*n^4+114*n^3+325*n^2+336*n+108)*x^5+(4*n^4+24*n^3+52*n^2+48*n+16)*x^6)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((300*n - 300 + (550*n^2 + 80*n - 630)*x + (300*n^3 + 640*n^2 - 472*n - 468)*x^2 + (55*n^4 + 351*n^3 + 173*n^2 - 407*n - 172)*x^3 + (3*n^5 + 53*n^4 + 113*n^3 + 3*n^2 - 124*n - 48)*x^4 + (2*n^5 + 10*n^4 + 14*n^3 - 2*n^2 - 16*n - 8)*x^5)*P[n - 1, x] - (125 + (375*n + 450)*x + (300*n^2 + 1200*n + 685)*x^2 + (75*n^3 + 800*n^2 + 1530*n + 594)*x^3 + (5*n^4 + 165*n^3 + 773*n^2 + 993*n + 326)*x^4 + (9*n^4 + 114*n^3 + 325*n^2 + 336*n + 108)*x^5 + (4*n^4 + 24*n^3 + 52*n^2 + 48*n + 16)*x^6)*D[P[n - 1, x], x])/(100*n - 100 + (125*n^2 - 65*n - 60)*x + (75*n^3 - 45*n^2 - 34*n + 4)*x^2 + (15*n^4 + 7*n^3 - 29*n^2 - 5*n + 12)*x^3 + (n^5 + n^4 - n^3 - n^2)*x^4)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((300*n - 300 + (550*n^2 + 80*n - 630)*x + (300*n^3 + 640*n^2 - 472*n - 468)*x^2 + (55*n^4 + 351*n^3 + 173*n^2 - 407*n - 172)*x^3 + (3*n^5 + 53*n^4 + 113*n^3 + 3*n^2 - 124*n - 48)*x^4 + (2*n^5 + 10*n^4 + 14*n^3 - 2*n^2 - 16*n - 8)*x^5)*v[i - 1] - (125 + (375*n + 450)*x + (300*n^2 + 1200*n + 685)*x^2 + (75*n^3 + 800*n^2 + 1530*n + 594)*x^3 + (5*n^4 + 165*n^3 + 773*n^2 + 993*n + 326)*x^4 + (9*n^4 + 114*n^3 + 325*n^2 + 336*n + 108)*x^5 + (4*n^4 + 24*n^3 + 52*n^2 + 48*n + 16)*x^6)*deriv(v[i - 1], x))/(100*n - 100 + (125*n^2 - 65*n - 60)*x + (75*n^3 - 45*n^2 - 34*n + 4)*x^2 + (15*n^4 + 7*n^3 - 29*n^2 - 5*n + 12)*x^3 + (n^5 + n^4 - n^3 - n^2)*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-299efe8218728fe3","steps":["(\u03b8+1)\u00b2"],"edges":["e-237624108b1e945b"],"text":"A245962 \u2192 (\u03b8+1)\u00b2"},"headline":"A245962 \u2192 (\u03b8+1)\u00b2","chain":null}