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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":"b77d086757d9c289fab723a260b8b6666a1e82b2683fe9c8cf25f40c7aeb4952","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-cca535f6115f5fc0","role":"sequence","name":"basis_map_A007318 of Triangle T(n,k), 0 <= k <= n, read by rows defined by: T(n,k) = (binomial(2*n,2*k) + binomial(2*n+1,2*k))/2.","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,       8;\n  2:     12,      56,       27;\n  3:     48,     320,      333,       64;\n  4:    192,    1664,     2664,     1216,      125;\n  5:    768,    8192,    17712,    13056,     3375,      216;\n  6:   3072,   38912,   105984,   108544,    46900,     7848,     343;\n  7:  12288,  180224,   592128,   778240,   477600,   137088,   16121,    512;\n  8:  49152,  819200,  3151872,  5062656,  4048000,  1679616,  345744,  30208,   729;\n  9: 196608, 3670016, 16183296, 30736384, 30368000, 16625664, 5019168, 780288, 52731, 1000;\n","recurrence_sentence":"The row polynomials satisfy -(48*n^3-96*n^2+60*n-12+(72*n^4-172*n^3+88*n^2+2*n-6)*x+(12*n^5-40*n^4-9*n^3+19*n^2-n-3)*x^2-(4*n^5+4*n^4+n^3)*x^3)*P(n,x) = -(192*n^3-384*n^2+240*n-48+(288*n^4-112*n^3-736*n^2+664*n-152)*x+(48*n^5+200*n^4-524*n^3-322*n^2+556*n-152)*x^2+(20*n^5-84*n^4-223*n^3+122*n-38)*x^3-(12*n^5+40*n^4+31*n^3-5*n^2-5*n-1)*x^4)*P(n-1,x) + ((96*n^2-96*n+24)*x+(144*n^3+112*n^2-256*n+76)*x^2+(24*n^4+160*n^3-62*n^2-226*n+70)*x^3+(16*n^4-16*n^3-116*n^2-80*n+16)*x^4-(8*n^4+32*n^3+38*n^2+14*n+2)*x^5)*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[(-(192*n^3 - 384*n^2 + 240*n - 48 + (288*n^4 - 112*n^3 - 736*n^2 + 664*n - 152)*x + (48*n^5 + 200*n^4 - 524*n^3 - 322*n^2 + 556*n - 152)*x^2 + (20*n^5 - 84*n^4 - 223*n^3 + 122*n - 38)*x^3 - (12*n^5 + 40*n^4 + 31*n^3 - 5*n^2 - 5*n - 1)*x^4)*P[n - 1, x] + ((96*n^2 - 96*n + 24)*x + (144*n^3 + 112*n^2 - 256*n + 76)*x^2 + (24*n^4 + 160*n^3 - 62*n^2 - 226*n + 70)*x^3 + (16*n^4 - 16*n^3 - 116*n^2 - 80*n + 16)*x^4 - (8*n^4 + 32*n^3 + 38*n^2 + 14*n + 2)*x^5)*D[P[n - 1, x], x])/(-48*n^3 + 96*n^2 - 60*n + 12 - (72*n^4 - 172*n^3 + 88*n^2 + 2*n - 6)*x - (12*n^5 - 40*n^4 - 9*n^3 + 19*n^2 - n - 3)*x^2 + (4*n^5 + 4*n^4 + n^3)*x^3)]];\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] = (-(192*n^3 - 384*n^2 + 240*n - 48 + (288*n^4 - 112*n^3 - 736*n^2 + 664*n - 152)*x + (48*n^5 + 200*n^4 - 524*n^3 - 322*n^2 + 556*n - 152)*x^2 + (20*n^5 - 84*n^4 - 223*n^3 + 122*n - 38)*x^3 - (12*n^5 + 40*n^4 + 31*n^3 - 5*n^2 - 5*n - 1)*x^4)*v[i - 1] + ((96*n^2 - 96*n + 24)*x + (144*n^3 + 112*n^2 - 256*n + 76)*x^2 + (24*n^4 + 160*n^3 - 62*n^2 - 226*n + 70)*x^3 + (16*n^4 - 16*n^3 - 116*n^2 - 80*n + 16)*x^4 - (8*n^4 + 32*n^3 + 38*n^2 + 14*n + 2)*x^5)*deriv(v[i - 1], x))/(-48*n^3 + 96*n^2 - 60*n + 12 - (72*n^4 - 172*n^3 + 88*n^2 + 2*n - 6)*x - (12*n^5 - 40*n^4 - 9*n^3 + 19*n^2 - n - 3)*x^2 + (4*n^5 + 4*n^4 + n^3)*x^3)); 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-c85abc56078671cc","steps":["basis A007318","(\u03b8+1)\u00b2"],"edges":["e-2b8d1605acbc4584","e-ee20aa398e43c4e6"],"text":"A232535 \u2192 basis A007318 \u2192 (\u03b8+1)\u00b2"},"headline":"A232535 \u2192 basis A007318 \u2192 (\u03b8+1)\u00b2","chain":null}