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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":"80fb967e08d4ae93b309b485238ecabb40102497c11341ee1b64dbca49f083d6","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-f268708db386b729","role":"sequence","name":"reversal of Triangle read by rows: T(n,k) = number of k-part order-consecutive partition of {1,2,...,n} (1 <= k <= n).","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:   2,     4;\n   3:   4,    20,     9;\n   4:   8,    64,    81,     16;\n   5:  16,   176,   369,    224,     25;\n   6:  32,   448,  1314,   1360,    500,     36;\n   7:  64,  1088,  4104,   6032,   3875,    972,     49;\n   8: 128,  2560, 11808,  22528,  20825,   9324,   1715,    64;\n   9: 256,  5888, 32112,  75392,  91225,  59472,  19894,  2816,   81;\n  10: 512, 13312, 83808, 233728, 350050, 300996, 148176, 38784, 4374, 100;\n","recurrence_sentence":"The row polynomials satisfy (16*n^3-16*n^2+(36*n^4-84*n^3+68*n^2-20*n)*x+(9*n^5+49*n^4-249*n^3+287*n^2-48*n-48)*x^2+(24*n^5-68*n^4-6*n^3+160*n^2-146*n+36)*x^3+(16*n^5-96*n^4+228*n^3-268*n^2+156*n-36)*x^4)*P(n,x) = (32*n^3-32*n^2+(72*n^4+24*n^3-184*n^2-40*n)*x+(18*n^5+278*n^4-474*n^3-318*n^2+256*n+240)*x^2+(75*n^5+269*n^4-1199*n^3+563*n^2+712*n-420)*x^3+(104*n^5-108*n^4-666*n^3+1304*n^2-798*n+180)*x^4+(48*n^5-192*n^4+236*n^3-76*n^2-16*n)*x^5)*P(n-1,x) - (32*n^2*x+(72*n^3+80*n^2-88*n-48)*x^2+(18*n^4+260*n^3-166*n^2-208*n+72)*x^3+(66*n^4+268*n^3-562*n^2+164*n+48)*x^4+(80*n^4+16*n^3-324*n^2+292*n-72)*x^5+(32*n^4-64*n^3+24*n^2+8*n)*x^6)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((32*n^3 - 32*n^2 + (72*n^4 + 24*n^3 - 184*n^2 - 40*n)*x + (18*n^5 + 278*n^4 - 474*n^3 - 318*n^2 + 256*n + 240)*x^2 + (75*n^5 + 269*n^4 - 1199*n^3 + 563*n^2 + 712*n - 420)*x^3 + (104*n^5 - 108*n^4 - 666*n^3 + 1304*n^2 - 798*n + 180)*x^4 + (48*n^5 - 192*n^4 + 236*n^3 - 76*n^2 - 16*n)*x^5)*P[n - 1, x] - (32*n^2*x + (72*n^3 + 80*n^2 - 88*n - 48)*x^2 + (18*n^4 + 260*n^3 - 166*n^2 - 208*n + 72)*x^3 + (66*n^4 + 268*n^3 - 562*n^2 + 164*n + 48)*x^4 + (80*n^4 + 16*n^3 - 324*n^2 + 292*n - 72)*x^5 + (32*n^4 - 64*n^3 + 24*n^2 + 8*n)*x^6)*D[P[n - 1, x], x])/(16*n^3 - 16*n^2 + (36*n^4 - 84*n^3 + 68*n^2 - 20*n)*x + (9*n^5 + 49*n^4 - 249*n^3 + 287*n^2 - 48*n - 48)*x^2 + (24*n^5 - 68*n^4 - 6*n^3 + 160*n^2 - 146*n + 36)*x^3 + (16*n^5 - 96*n^4 + 228*n^3 - 268*n^2 + 156*n - 36)*x^4)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((32*n^3 - 32*n^2 + (72*n^4 + 24*n^3 - 184*n^2 - 40*n)*x + (18*n^5 + 278*n^4 - 474*n^3 - 318*n^2 + 256*n + 240)*x^2 + (75*n^5 + 269*n^4 - 1199*n^3 + 563*n^2 + 712*n - 420)*x^3 + (104*n^5 - 108*n^4 - 666*n^3 + 1304*n^2 - 798*n + 180)*x^4 + (48*n^5 - 192*n^4 + 236*n^3 - 76*n^2 - 16*n)*x^5)*v[i - 1] - (32*n^2*x + (72*n^3 + 80*n^2 - 88*n - 48)*x^2 + (18*n^4 + 260*n^3 - 166*n^2 - 208*n + 72)*x^3 + (66*n^4 + 268*n^3 - 562*n^2 + 164*n + 48)*x^4 + (80*n^4 + 16*n^3 - 324*n^2 + 292*n - 72)*x^5 + (32*n^4 - 64*n^3 + 24*n^2 + 8*n)*x^6)*deriv(v[i - 1], x))/(16*n^3 - 16*n^2 + (36*n^4 - 84*n^3 + 68*n^2 - 20*n)*x + (9*n^5 + 49*n^4 - 249*n^3 + 287*n^2 - 48*n - 48)*x^2 + (24*n^5 - 68*n^4 - 6*n^3 + 160*n^2 - 146*n + 36)*x^3 + (16*n^5 - 96*n^4 + 228*n^3 - 268*n^2 + 156*n - 36)*x^4)); 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-2821c21537bbbcdd","steps":["reverse","(\u03b8+1)\u00b2"],"edges":["e-23a5df9ffca078bb","e-e70ae255d0b8ff6a"],"text":"A056242 \u2192 reverse \u2192 (\u03b8+1)\u00b2"},"headline":"A056242 \u2192 reverse \u2192 (\u03b8+1)\u00b2","chain":null}