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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":"834bb11df87a30a182c8165a8d992d4b4e90347c36ddd6fc30c66a840e6782cf","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-bb74f03367de3ec8","role":"sequence","name":"Generalized Narayana numbers, T(n, k) = 3/(n + 1)*binomial(n + 1, k + 2)*binomial(n + 1, k - 1), triangular array read by rows.","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:  3,   12;\n  2:  6,   60,     54;\n  3: 10,  180,    405,    160;\n  4: 15,  420,   1701,   1680,     375;\n  5: 21,  840,   5292,   9408,    5250,     756;\n  6: 28, 1512,  13608,  37632,   37800,   13608,    1372;\n  7: 36, 2520,  30618, 120960,  189000,  122472,   30870,   2304;\n  8: 45, 3960,  62370, 332640,  742500,  748440,  339570,  63360,   3645;\n  9: 55, 5940, 117612, 813120, 2450250, 3528360, 2490180, 836352, 120285, 5500;\n","recurrence_sentence":"The row polynomials satisfy (4*n^2+24*n+(5*n^4+41*n^3+60*n^2-36*n)*x+(n^6+12*n^5+45*n^4+54*n^3)*x^2)*P(n,x) = (4*n^2+32*n+48+(5*n^4+51*n^3+190*n^2+244*n+56)*x+(n^6+14*n^5+93*n^4+304*n^3+444*n^2+256*n+64)*x^2+(3*n^6+36*n^5+164*n^4+363*n^3+430*n^2+276*n+72)*x^3)*P(n-1,x) + ((8*n+16)*x+(10*n^3+57*n^2+80*n+12)*x^2+(2*n^5+22*n^4+78*n^3+98*n^2+32*n+8)*x^3-(2*n^5+22*n^4+88*n^3+155*n^2+120*n+36)*x^4)*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*n^2 + 32*n + 48 + (5*n^4 + 51*n^3 + 190*n^2 + 244*n + 56)*x + (n^6 + 14*n^5 + 93*n^4 + 304*n^3 + 444*n^2 + 256*n + 64)*x^2 + (3*n^6 + 36*n^5 + 164*n^4 + 363*n^3 + 430*n^2 + 276*n + 72)*x^3)*P[n - 1, x] + ((8*n + 16)*x + (10*n^3 + 57*n^2 + 80*n + 12)*x^2 + (2*n^5 + 22*n^4 + 78*n^3 + 98*n^2 + 32*n + 8)*x^3 - (2*n^5 + 22*n^4 + 88*n^3 + 155*n^2 + 120*n + 36)*x^4)*D[P[n - 1, x], x])/(4*n^2 + 24*n + (5*n^4 + 41*n^3 + 60*n^2 - 36*n)*x + (n^6 + 12*n^5 + 45*n^4 + 54*n^3)*x^2)]];\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*n^2 + 32*n + 48 + (5*n^4 + 51*n^3 + 190*n^2 + 244*n + 56)*x + (n^6 + 14*n^5 + 93*n^4 + 304*n^3 + 444*n^2 + 256*n + 64)*x^2 + (3*n^6 + 36*n^5 + 164*n^4 + 363*n^3 + 430*n^2 + 276*n + 72)*x^3)*v[i - 1] + ((8*n + 16)*x + (10*n^3 + 57*n^2 + 80*n + 12)*x^2 + (2*n^5 + 22*n^4 + 78*n^3 + 98*n^2 + 32*n + 8)*x^3 - (2*n^5 + 22*n^4 + 88*n^3 + 155*n^2 + 120*n + 36)*x^4)*deriv(v[i - 1], x))/(4*n^2 + 24*n + (5*n^4 + 41*n^3 + 60*n^2 - 36*n)*x + (n^6 + 12*n^5 + 45*n^4 + 54*n^3)*x^2)); 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-bb74f03367de3ec8","steps":["(\u03b8+1)\u00b2"],"edges":["e-b52fd312b43db576"],"text":"A145597 \u2192 (\u03b8+1)\u00b2"},"headline":"A145597 \u2192 (\u03b8+1)\u00b2","chain":null}