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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":"309a12816e4c17112dcceb8edecb05a169b1ff8009bc98977bd2ca600b6757f8","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-67b93749ab954f15","role":"sequence","name":"Triangle read by rows: T(n,k) is the number of end rhyme patterns of a poem of an even number of lines (2n) with 1<=k<=n evenly rhymed sounds.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1: 1;\n   2: 1,      6;\n   3: 1,     30,        45;\n   4: 1,    126,       630,         420;\n   5: 1,    510,      6615,       12600,         4725;\n   6: 1,   2046,     63360,      263340,       259875,        62370;\n   7: 1,   8190,    585585,     4804800,      9459450,      5675670,       945945;\n   8: 1,  32766,   5331690,    82342260,    290540250,    329188860,    132432300,     16216200;\n   9: 1, 131070,  48230955,  1367238600,   8189856675,  15685930260,  11481880410,   3308104800,   310134825;\n  10: 1, 524286, 435061620, 22325973180, 220127851125, 672799597470, 800515415700, 408550942800, 88388425125, 6547290750;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -3*x^2*P(n-2,x) - (2*x^2+2*x^3)*d/dx P(n-2,x) - x^3*d^2/dx^2 P(n-2,x) + (1+6*x)*P(n-1,x) + (3*x+2*x^2)*d/dx P(n-1,x) + x^2*d^2/dx^2 P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + 6*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 1 + 6*x;\nP[n_, x_] := P[n, x] = Expand[-3*x^2*P[n - 2, x] - (2*x^2 + 2*x^3)*D[P[n - 2, x], x] - x^3*D[P[n - 2, x], {x, 2}] + (1 + 6*x)*P[n - 1, x] + (3*x + 2*x^2)*D[P[n - 1, x], x] + x^2*D[P[n - 1, x], {x, 2}]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + 6*x; for(n = 3, N, my(i = n); v[i] = -3*x^2*v[i - 2] - (2*x^2 + 2*x^3)*deriv(v[i - 2], x) - x^3*deriv(deriv(v[i - 2], x), x) + (1 + 6*x)*v[i - 1] + (3*x + 2*x^2)*deriv(v[i - 1], x) + x^2*deriv(deriv(v[i - 1], x), x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-67b93749ab954f15","steps":["\u03b8+1"],"edges":["e-03c8bde6cb3e7208"],"text":"A156289 \u2192 \u03b8+1"},"headline":"A156289 \u2192 \u03b8+1","chain":null}