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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":"570bf8b845084e545f43ea41843bc6d67a96bb7e2f686494c332f2422f85cd2e","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-a6f3b35778192ffb","role":"sequence","name":"Triangle T(n,k) = number of k-part order-consecutive partitions of n (1<=k<=n).","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,  171,   160,    25;\n  5: 1,  60,  405,   720,   375,     36;\n  6: 1,  84,  810,  2256,  2250,    756,    49;\n  7: 1, 112, 1449,  5712,  8925,   5796,  1372,    64;\n  8: 1, 144, 2394, 12544, 27675,  28224, 13034,  2304,   81;\n  9: 1, 180, 3726, 24864, 72675, 104652, 76146, 26496, 3645, 100;\n","recurrence_sentence":"The row polynomials satisfy (36-(22*n^2-130*n+100)*x+(n^4-14*n^3+193*n^2-304*n+40)*x^2-(8*n^4-88*n^3+146*n^2-38*n-4)*x^3+(16*n^4-32*n^3-4*n^2+28*n-4)*x^4)*P(n,x) = -(36-(22*n^2-86*n-44)*x+(n^4-10*n^3+135*n^2+130*n-40)*x^2-(7*n^4-46*n^3-205*n^2-108*n+100)*x^3+(9*n^4+78*n^3+223*n^2+10*n-104)*x^4+(8*n^4+88*n^3+66*n^2-34*n-20)*x^5+(16*n^4+32*n^3-4*n^2-12*n+4)*x^6)*P(n-2,x) + (72-(44*n^2-216*n-88)*x+(2*n^4-24*n^3+290*n^2+44*n-472)*x^2-(14*n^4-120*n^3-378*n^2+516*n+472)*x^3+(16*n^4+144*n^3-172*n^2-168*n+88)*x^4+(32*n^4-104*n^2+16*n+72)*x^5)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 4*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 4*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(36 - (22*n^2 - 86*n - 44)*x + (n^4 - 10*n^3 + 135*n^2 + 130*n - 40)*x^2 - (7*n^4 - 46*n^3 - 205*n^2 - 108*n + 100)*x^3 + (9*n^4 + 78*n^3 + 223*n^2 + 10*n - 104)*x^4 + (8*n^4 + 88*n^3 + 66*n^2 - 34*n - 20)*x^5 + (16*n^4 + 32*n^3 - 4*n^2 - 12*n + 4)*x^6)*P[n - 2, x] + (72 - (44*n^2 - 216*n - 88)*x + (2*n^4 - 24*n^3 + 290*n^2 + 44*n - 472)*x^2 - (14*n^4 - 120*n^3 - 378*n^2 + 516*n + 472)*x^3 + (16*n^4 + 144*n^3 - 172*n^2 - 168*n + 88)*x^4 + (32*n^4 - 104*n^2 + 16*n + 72)*x^5)*P[n - 1, x])/(36 - (22*n^2 - 130*n + 100)*x + (n^4 - 14*n^3 + 193*n^2 - 304*n + 40)*x^2 - (8*n^4 - 88*n^3 + 146*n^2 - 38*n - 4)*x^3 + (16*n^4 - 32*n^3 - 4*n^2 + 28*n - 4)*x^4)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (-(36 - (22*n^2 - 86*n - 44)*x + (n^4 - 10*n^3 + 135*n^2 + 130*n - 40)*x^2 - (7*n^4 - 46*n^3 - 205*n^2 - 108*n + 100)*x^3 + (9*n^4 + 78*n^3 + 223*n^2 + 10*n - 104)*x^4 + (8*n^4 + 88*n^3 + 66*n^2 - 34*n - 20)*x^5 + (16*n^4 + 32*n^3 - 4*n^2 - 12*n + 4)*x^6)*v[i - 2] + (72 - (44*n^2 - 216*n - 88)*x + (2*n^4 - 24*n^3 + 290*n^2 + 44*n - 472)*x^2 - (14*n^4 - 120*n^3 - 378*n^2 + 516*n + 472)*x^3 + (16*n^4 + 144*n^3 - 172*n^2 - 168*n + 88)*x^4 + (32*n^4 - 104*n^2 + 16*n + 72)*x^5)*v[i - 1])/(36 - (22*n^2 - 130*n + 100)*x + (n^4 - 14*n^3 + 193*n^2 - 304*n + 40)*x^2 - (8*n^4 - 88*n^3 + 146*n^2 - 38*n - 4)*x^3 + (16*n^4 - 32*n^3 - 4*n^2 + 28*n - 4)*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-a6f3b35778192ffb","steps":["(\u03b8+1)\u00b2"],"edges":["e-9c896b6e10f8a55d"],"text":"A056241 \u2192 (\u03b8+1)\u00b2"},"headline":"A056241 \u2192 (\u03b8+1)\u00b2","chain":null}