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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":"ea81cdc7efabee4bdb8e2fb14545b0952cbc5ee679704057e5f35342ace1b381","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-0bad942bfffd4f03","role":"sequence","name":"basis_map_A007318 of Pascal-(1,4,1) array.","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,      4;\n  2:     8,     32,       9;\n  3:    24,    144,     126,      16;\n  4:    80,    640,     900,     320,      25;\n  5:   256,   2560,    5040,    3200,     650,      36;\n  6:   832,   9984,   25200,   23040,    8400,    1152,     49;\n  7:  2688,  37632,  116064,  139520,   75400,   18288,   1862,    64;\n  8:  8704, 139264,  507456,  756736,  546400,  198144,  35084,  2816,   81;\n  9: 28160, 506880, 2131200, 3799040, 3448800, 1690560, 446880, 61440, 4050, 100;\n","recurrence_sentence":"The row polynomials satisfy (80+(240*n-40)*x+(220*n^2-20*n-28)*x^2+(60*n^3+50*n^2-34*n-14)*x^3+(5*n^4+10*n^3-5*n^2-4*n+2)*x^4-(n^3+n^2)*x^5)*P(n,x) = (320+(960*n+1120)*x+(880*n^2+2640*n+1488)*x^2+(240*n^3+1800*n^2+2664*n+936)*x^3+(20*n^4+360*n^3+1140*n^2+1128*n+280)*x^4+(20*n^4+116*n^3+204*n^2+124*n+24)*x^5-(4*n^3+16*n^2+20*n+8)*x^6)*P(n-2,x) + (160+(480*n+240)*x+(440*n^2+640*n+64)*x^2+(120*n^3+500*n^2+312*n-120)*x^3+(10*n^4+100*n^3+180*n^2+28*n-54)*x^4+(5*n^4+18*n^3+10*n^2-4*n-1)*x^5-(n^3+3*n^2+3*n+1)*x^6)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 4*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + 4*x;\nP[n_, x_] := P[n, x] = Expand[Together[((320 + (960*n + 1120)*x + (880*n^2 + 2640*n + 1488)*x^2 + (240*n^3 + 1800*n^2 + 2664*n + 936)*x^3 + (20*n^4 + 360*n^3 + 1140*n^2 + 1128*n + 280)*x^4 + (20*n^4 + 116*n^3 + 204*n^2 + 124*n + 24)*x^5 - (4*n^3 + 16*n^2 + 20*n + 8)*x^6)*P[n - 2, x] + (160 + (480*n + 240)*x + (440*n^2 + 640*n + 64)*x^2 + (120*n^3 + 500*n^2 + 312*n - 120)*x^3 + (10*n^4 + 100*n^3 + 180*n^2 + 28*n - 54)*x^4 + (5*n^4 + 18*n^3 + 10*n^2 - 4*n - 1)*x^5 - (n^3 + 3*n^2 + 3*n + 1)*x^6)*P[n - 1, x])/(80 + (240*n - 40)*x + (220*n^2 - 20*n - 28)*x^2 + (60*n^3 + 50*n^2 - 34*n - 14)*x^3 + (5*n^4 + 10*n^3 - 5*n^2 - 4*n + 2)*x^4 - (n^3 + n^2)*x^5)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = ((320 + (960*n + 1120)*x + (880*n^2 + 2640*n + 1488)*x^2 + (240*n^3 + 1800*n^2 + 2664*n + 936)*x^3 + (20*n^4 + 360*n^3 + 1140*n^2 + 1128*n + 280)*x^4 + (20*n^4 + 116*n^3 + 204*n^2 + 124*n + 24)*x^5 - (4*n^3 + 16*n^2 + 20*n + 8)*x^6)*v[i - 2] + (160 + (480*n + 240)*x + (440*n^2 + 640*n + 64)*x^2 + (120*n^3 + 500*n^2 + 312*n - 120)*x^3 + (10*n^4 + 100*n^3 + 180*n^2 + 28*n - 54)*x^4 + (5*n^4 + 18*n^3 + 10*n^2 - 4*n - 1)*x^5 - (n^3 + 3*n^2 + 3*n + 1)*x^6)*v[i - 1])/(80 + (240*n - 40)*x + (220*n^2 - 20*n - 28)*x^2 + (60*n^3 + 50*n^2 - 34*n - 14)*x^3 + (5*n^4 + 10*n^3 - 5*n^2 - 4*n + 2)*x^4 - (n^3 + n^2)*x^5)); 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-ed9434bfd889c409","steps":["x \u21a6 x+1","(\u03b8+1)\u00b2"],"edges":["e-0f7951445df45664","e-bdc48048b1248ea2"],"text":"A081579 \u2192 x \u21a6 x+1 \u2192 (\u03b8+1)\u00b2"},"headline":"A081579 \u2192 x \u21a6 x+1 \u2192 (\u03b8+1)\u00b2","chain":null}