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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":"f2d15a7f5187c2043f4f7119f08894219b5dced8199a0f25e34f4046ccccc513","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-1f449c58b1930aa5","role":"sequence","name":"basis_map_A013611 of A convolution triangle of numbers based on A001906 (even-indexed Fibonacci numbers).","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,      16;\n  2:     15,     128,      144;\n  3:     56,     736,     1728,      1024;\n  4:    209,    3712,    13392,     16384,      6400;\n  5:    780,   17456,    85248,    159744,    128000,      36864;\n  6:   2911,   78592,   484704,   1228800,   1504000,     884736,     200704;\n  7:  10864,  343488,  2560896,   8202240,  13568000,   12165120,    5619712,   1048576;\n  8:  40545, 1469184, 12850272,  49790976, 104032000,  125927424,   88510464,  33554432,   5308416;\n  9: 151316, 6181968, 62062848, 282341376, 713753600, 1090768896, 1034027008, 595591168, 191102976, 26214400;\n","recurrence_sentence":"The row polynomials satisfy (72*n+24*n*x-(240*n^3-32*n)*x^2-160*n^3*x^3+32*n^5*x^4)*P(n,x) = (144*n+144+(252*n^2+624*n+372)*x-(480*n^3+528*n^2-256*n-304)*x^2-(96*n^4+1136*n^3+1856*n^2+848*n+32)*x^3+(64*n^5+64*n^4-576*n^3-1152*n^2-704*n-128)*x^4+(64*n^5+192*n^4+192*n^3+64*n^2)*x^5)*P(n-1,x) + (27+108*x-(252*n^2+252*n-108)*x^2-(912*n^2+912*n+64)*x^3+(48*n^4+96*n^3-928*n^2-976*n-160)*x^4+(128*n^4+256*n^3-192*n^2-320*n-64)*x^5+(64*n^4+128*n^3+64*n^2)*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[((144*n + 144 + (252*n^2 + 624*n + 372)*x - (480*n^3 + 528*n^2 - 256*n - 304)*x^2 - (96*n^4 + 1136*n^3 + 1856*n^2 + 848*n + 32)*x^3 + (64*n^5 + 64*n^4 - 576*n^3 - 1152*n^2 - 704*n - 128)*x^4 + (64*n^5 + 192*n^4 + 192*n^3 + 64*n^2)*x^5)*P[n - 1, x] + (27 + 108*x - (252*n^2 + 252*n - 108)*x^2 - (912*n^2 + 912*n + 64)*x^3 + (48*n^4 + 96*n^3 - 928*n^2 - 976*n - 160)*x^4 + (128*n^4 + 256*n^3 - 192*n^2 - 320*n - 64)*x^5 + (64*n^4 + 128*n^3 + 64*n^2)*x^6)*D[P[n - 1, x], x])/(72*n + 24*n*x - (240*n^3 - 32*n)*x^2 - 160*n^3*x^3 + 32*n^5*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] = ((144*n + 144 + (252*n^2 + 624*n + 372)*x - (480*n^3 + 528*n^2 - 256*n - 304)*x^2 - (96*n^4 + 1136*n^3 + 1856*n^2 + 848*n + 32)*x^3 + (64*n^5 + 64*n^4 - 576*n^3 - 1152*n^2 - 704*n - 128)*x^4 + (64*n^5 + 192*n^4 + 192*n^3 + 64*n^2)*x^5)*v[i - 1] + (27 + 108*x - (252*n^2 + 252*n - 108)*x^2 - (912*n^2 + 912*n + 64)*x^3 + (48*n^4 + 96*n^3 - 928*n^2 - 976*n - 160)*x^4 + (128*n^4 + 256*n^3 - 192*n^2 - 320*n - 64)*x^5 + (64*n^4 + 128*n^3 + 64*n^2)*x^6)*deriv(v[i - 1], x))/(72*n + 24*n*x - (240*n^3 - 32*n)*x^2 - 160*n^3*x^3 + 32*n^5*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-f287bed89e07609f","steps":["basis A013611","(\u03b8+1)\u00b2"],"edges":["e-362ae2b737f320ab","e-d5d7ea12be34f1ea"],"text":"A125662 \u2192 basis A013611 \u2192 (\u03b8+1)\u00b2"},"headline":"A125662 \u2192 basis A013611 \u2192 (\u03b8+1)\u00b2","chain":null}