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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":"e64213c66bee8f23e81cbdf7ad295b191c0a0f2b1506b2ffb2e9122485cbbc12","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-b6cda967a092feb1","role":"sequence","name":"basis_map_A007318 of Triangle read by rows: T(n,k) is the number of double rise-bicolored Dyck paths (double rises come in two colors; also called marked Dyck paths) of semilength n and having k double rises (0 <= k <= n-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:      1;\n  1:      3,        8;\n  2:     11,       56,       36;\n  3:     45,      336,      432,       128;\n  4:    197,     1936,     3744,      2304,       400;\n  5:    903,    11000,    28440,     26880,     10000,      1152;\n  6:   4279,    62184,   201420,    257920,    150000,     38016,     3136;\n  7:  20793,   351008,  1366848,   2213120,   1764000,    709632,   131712,     8192;\n  8: 103049,  1981472,  9015552,  17669120,  17953600,   9999360,  2985472,   425984,   20736;\n  9: 518859, 11194344, 58282416, 134163456, 166017600, 118734336, 49523712, 11501568, 1306368, 51200;\n","recurrence_sentence":"The row polynomials satisfy (2*n+4+(10*n^2+27*n+14)*x+(18*n^3+54*n^2+40*n+8)*x^2+(12*n^4+43*n^3+41*n^2+2*n-8)*x^3+(2*n^5+8*n^4+10*n^3+4*n^2)*x^4)*P(n,x) = (14*n+4+(84*n^2+103*n+14)*x+(150*n^3+390*n^2+220*n+8)*x^2+(92*n^4+443*n^3+557*n^2+146*n-8)*x^3+(14*n^5+152*n^4+362*n^3+256*n^2+8*n)*x^4+(12*n^5+56*n^4+80*n^3+36*n^2+8*n)*x^5)*P(n-1,x) - (1+(12*n+12)*x+(30*n^2+91*n+49)*x^2+(24*n^3+164*n^2+256*n+92)*x^3+(4*n^4+92*n^3+314*n^2+329*n+86)*x^4+(12*n^4+108*n^3+244*n^2+188*n+40)*x^5+(8*n^4+40*n^3+64*n^2+36*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[((14*n + 4 + (84*n^2 + 103*n + 14)*x + (150*n^3 + 390*n^2 + 220*n + 8)*x^2 + (92*n^4 + 443*n^3 + 557*n^2 + 146*n - 8)*x^3 + (14*n^5 + 152*n^4 + 362*n^3 + 256*n^2 + 8*n)*x^4 + (12*n^5 + 56*n^4 + 80*n^3 + 36*n^2 + 8*n)*x^5)*P[n - 1, x] - (1 + (12*n + 12)*x + (30*n^2 + 91*n + 49)*x^2 + (24*n^3 + 164*n^2 + 256*n + 92)*x^3 + (4*n^4 + 92*n^3 + 314*n^2 + 329*n + 86)*x^4 + (12*n^4 + 108*n^3 + 244*n^2 + 188*n + 40)*x^5 + (8*n^4 + 40*n^3 + 64*n^2 + 36*n + 8)*x^6)*D[P[n - 1, x], x])/(2*n + 4 + (10*n^2 + 27*n + 14)*x + (18*n^3 + 54*n^2 + 40*n + 8)*x^2 + (12*n^4 + 43*n^3 + 41*n^2 + 2*n - 8)*x^3 + (2*n^5 + 8*n^4 + 10*n^3 + 4*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; for(n = 1, N, my(i = n + 1); v[i] = ((14*n + 4 + (84*n^2 + 103*n + 14)*x + (150*n^3 + 390*n^2 + 220*n + 8)*x^2 + (92*n^4 + 443*n^3 + 557*n^2 + 146*n - 8)*x^3 + (14*n^5 + 152*n^4 + 362*n^3 + 256*n^2 + 8*n)*x^4 + (12*n^5 + 56*n^4 + 80*n^3 + 36*n^2 + 8*n)*x^5)*v[i - 1] - (1 + (12*n + 12)*x + (30*n^2 + 91*n + 49)*x^2 + (24*n^3 + 164*n^2 + 256*n + 92)*x^3 + (4*n^4 + 92*n^3 + 314*n^2 + 329*n + 86)*x^4 + (12*n^4 + 108*n^3 + 244*n^2 + 188*n + 40)*x^5 + (8*n^4 + 40*n^3 + 64*n^2 + 36*n + 8)*x^6)*deriv(v[i - 1], x))/(2*n + 4 + (10*n^2 + 27*n + 14)*x + (18*n^3 + 54*n^2 + 40*n + 8)*x^2 + (12*n^4 + 43*n^3 + 41*n^2 + 2*n - 8)*x^3 + (2*n^5 + 8*n^4 + 10*n^3 + 4*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-8ce61d302cec295b","steps":["basis A007318","(\u03b8+1)\u00b2"],"edges":["e-6804282af6df94ec","e-8d02ad89cd68e20f"],"text":"A114687 \u2192 basis A007318 \u2192 (\u03b8+1)\u00b2"},"headline":"A114687 \u2192 basis A007318 \u2192 (\u03b8+1)\u00b2","chain":null}