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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":"2b482e876d4c00c5af295c6bbde5d7e4c61be975c18b3ad69685da32224cc7b5","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-1ff7b1d8c09df87e","role":"sequence","name":"derivative of Triangle read by rows: T(n,k) is the number of lattice paths from (0,0) to (n,n) using steps E=(1,0), N=(0,1) and D=(1,1) (i.e., bilateral Schroeder paths), having k D=(1,1) steps.","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:      0;\n   2:      1;\n   3:      6,       8;\n   4:     30,      96,       27;\n   5:    140,     720,      540,       64;\n   6:    630,    4480,     5670,     1920,      125;\n   7:   2772,   25200,    45360,    26880,     5250,     216;\n   8:  12012,  133056,   311850,   268800,    94500,   12096,    343;\n   9:  51480,  672672,  1945944,  2217600,  1155000,  272160,  24696,   512;\n  10: 218790, 3294720, 11351340, 16144128, 11261250, 3991680, 679140, 46080, 729;\n","recurrence_sentence":"The row polynomials satisfy (4*n^4-28*n^3+73*n^2-84*n+36+(6*n^5-60*n^4+238*n^3-468*n^2+456*n-176)*x+(n^6-12*n^5+60*n^4-160*n^3+240*n^2-192*n+64)*x^2)*P(n,x) = (16*n^4-104*n^3+252*n^2-270*n+108+(24*n^5-180*n^4+520*n^3-728*n^2+496*n-132)*x+(4*n^6-16*n^5-43*n^4+338*n^3-701*n^2+630*n-214)*x^2+(3*n^6-28*n^5+104*n^4-196*n^3+197*n^2-100*n+20)*x^3)*P(n-1,x) - ((8*n^3-36*n^2+54*n-27)*x+(12*n^4-64*n^3+124*n^2-104*n+32)*x^2+(2*n^5-n^4-40*n^3+123*n^2-138*n+55)*x^3+(2*n^5-13*n^4+32*n^3-37*n^2+20*n-4)*x^4)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 1.","mathematica":"P[1, x_] := 0;\nP[2, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((16*n^4 - 104*n^3 + 252*n^2 - 270*n + 108 + (24*n^5 - 180*n^4 + 520*n^3 - 728*n^2 + 496*n - 132)*x + (4*n^6 - 16*n^5 - 43*n^4 + 338*n^3 - 701*n^2 + 630*n - 214)*x^2 + (3*n^6 - 28*n^5 + 104*n^4 - 196*n^3 + 197*n^2 - 100*n + 20)*x^3)*P[n - 1, x] - ((8*n^3 - 36*n^2 + 54*n - 27)*x + (12*n^4 - 64*n^3 + 124*n^2 - 104*n + 32)*x^2 + (2*n^5 - n^4 - 40*n^3 + 123*n^2 - 138*n + 55)*x^3 + (2*n^5 - 13*n^4 + 32*n^3 - 37*n^2 + 20*n - 4)*x^4)*D[P[n - 1, x], x])/(4*n^4 - 28*n^3 + 73*n^2 - 84*n + 36 + (6*n^5 - 60*n^4 + 238*n^3 - 468*n^2 + 456*n - 176)*x + (n^6 - 12*n^5 + 60*n^4 - 160*n^3 + 240*n^2 - 192*n + 64)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 1; for(n = 3, N, my(i = n); v[i] = ((16*n^4 - 104*n^3 + 252*n^2 - 270*n + 108 + (24*n^5 - 180*n^4 + 520*n^3 - 728*n^2 + 496*n - 132)*x + (4*n^6 - 16*n^5 - 43*n^4 + 338*n^3 - 701*n^2 + 630*n - 214)*x^2 + (3*n^6 - 28*n^5 + 104*n^4 - 196*n^3 + 197*n^2 - 100*n + 20)*x^3)*v[i - 1] - ((8*n^3 - 36*n^2 + 54*n - 27)*x + (12*n^4 - 64*n^3 + 124*n^2 - 104*n + 32)*x^2 + (2*n^5 - n^4 - 40*n^3 + 123*n^2 - 138*n + 55)*x^3 + (2*n^5 - 13*n^4 + 32*n^3 - 37*n^2 + 20*n - 4)*x^4)*deriv(v[i - 1], x))/(4*n^4 - 28*n^3 + 73*n^2 - 84*n + 36 + (6*n^5 - 60*n^4 + 238*n^3 - 468*n^2 + 456*n - 176)*x + (n^6 - 12*n^5 + 60*n^4 - 160*n^3 + 240*n^2 - 192*n + 64)*x^2)); 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-78ba05b4a3dd925d","steps":["D","(\u03b8+1)\u00b2"],"edges":["e-4c8816dc3f762029","e-720f8bf95ddad7e2"],"text":"A104684 \u2192 D \u2192 (\u03b8+1)\u00b2"},"headline":"A104684 \u2192 D \u2192 (\u03b8+1)\u00b2","chain":null}