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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":"f1f3ad4510f1359897b78e2550744723568af49de2e601bc849a404306c833a4","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-07e5356ad37f61ea","role":"sequence","name":"Triangle read by rows: T(n,k) is the number of k-matchings of the corona K'(n) of the complete graph K(n) and the complete graph K(1); in other words, K'(n) is the graph constructed from K(n) by adding for each vertex v a new vertex v' and the edge vv'.","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,   4;\n  2: 1,  12,    9;\n  3: 1,  24,   54,    16;\n  4: 1,  40,  189,   160,     25;\n  5: 1,  60,  495,   880,    375,     36;\n  6: 1,  84, 1080,  3440,   3000,    756,     49;\n  7: 1, 112, 2079, 10640,  16625,   8316,   1372,    64;\n  8: 1, 144, 3654, 27776,  70875,  62496,  19894,  2304,   81;\n  9: 1, 180, 5994, 63840, 247275, 356076, 195510, 42624, 3645, 100;\n","recurrence_sentence":"The row polynomials satisfy -(4+(6*n^2+10*n-20)*x+(n^4+6*n^3-n^2-54*n+28)*x^2-(n^4+14*n^3+43*n^2-30*n+16)*x^3+(4*n^3+30*n^2+2*n)*x^4-4*n^2*x^5)*P(n,x) = -(4+(6*n^2+26*n-20)*x+(n^4+16*n^3+33*n^2-78*n-8)*x^2+(n^5+9*n^4+11*n^3-85*n^2-48*n-12)*x^3-(5*n^4+40*n^3+57*n^2+30*n+8)*x^4+(8*n^3+42*n^2+62*n+28)*x^5-(4*n^2+8*n+4)*x^6)*P(n-1,x) + (8*x^2+(12*n^2+44*n)*x^3+(2*n^4+20*n^3+58*n^2-8)*x^4-(8*n^3+60*n^2+76*n+32)*x^5+(8*n^2+16*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[(-(4 + (6*n^2 + 26*n - 20)*x + (n^4 + 16*n^3 + 33*n^2 - 78*n - 8)*x^2 + (n^5 + 9*n^4 + 11*n^3 - 85*n^2 - 48*n - 12)*x^3 - (5*n^4 + 40*n^3 + 57*n^2 + 30*n + 8)*x^4 + (8*n^3 + 42*n^2 + 62*n + 28)*x^5 - (4*n^2 + 8*n + 4)*x^6)*P[n - 1, x] + (8*x^2 + (12*n^2 + 44*n)*x^3 + (2*n^4 + 20*n^3 + 58*n^2 - 8)*x^4 - (8*n^3 + 60*n^2 + 76*n + 32)*x^5 + (8*n^2 + 16*n + 8)*x^6)*D[P[n - 1, x], x])/(-4 - (6*n^2 + 10*n - 20)*x - (n^4 + 6*n^3 - n^2 - 54*n + 28)*x^2 + (n^4 + 14*n^3 + 43*n^2 - 30*n + 16)*x^3 - (4*n^3 + 30*n^2 + 2*n)*x^4 + 4*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; for(n = 1, N, my(i = n + 1); v[i] = (-(4 + (6*n^2 + 26*n - 20)*x + (n^4 + 16*n^3 + 33*n^2 - 78*n - 8)*x^2 + (n^5 + 9*n^4 + 11*n^3 - 85*n^2 - 48*n - 12)*x^3 - (5*n^4 + 40*n^3 + 57*n^2 + 30*n + 8)*x^4 + (8*n^3 + 42*n^2 + 62*n + 28)*x^5 - (4*n^2 + 8*n + 4)*x^6)*v[i - 1] + (8*x^2 + (12*n^2 + 44*n)*x^3 + (2*n^4 + 20*n^3 + 58*n^2 - 8)*x^4 - (8*n^3 + 60*n^2 + 76*n + 32)*x^5 + (8*n^2 + 16*n + 8)*x^6)*deriv(v[i - 1], x))/(-4 - (6*n^2 + 10*n - 20)*x - (n^4 + 6*n^3 - n^2 - 54*n + 28)*x^2 + (n^4 + 14*n^3 + 43*n^2 - 30*n + 16)*x^3 - (4*n^3 + 30*n^2 + 2*n)*x^4 + 4*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-07e5356ad37f61ea","steps":["(\u03b8+1)\u00b2"],"edges":["e-46744df2075291bc"],"text":"A100862 \u2192 (\u03b8+1)\u00b2"},"headline":"A100862 \u2192 (\u03b8+1)\u00b2","chain":null}