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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":"acc6f553960765889519e67ba0b38c991368029963c115011cf42161d62c072f","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-d6b7687ba0ff59dd","role":"sequence","name":"Riordan array ((1+x)/(1-2x),x(1+x)/(1-2x)).","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,     4;\n  2:   6,    24,      9;\n  3:  12,    84,     81,     16;\n  4:  24,   240,    405,    192,     25;\n  5:  48,   624,   1539,   1248,    375,     36;\n  6:  96,  1536,   5022,   5952,   3000,    648,    49;\n  7: 192,  3648,  14904,  23568,  17250,   6156,  1029,    64;\n  8: 384,  8448,  41472,  82560,  80625,  41472, 11319,  1536,   81;\n  9: 768, 19200, 110160, 265344, 327075, 223884, 87465, 19200, 2187, 100;\n","recurrence_sentence":"The row polynomials satisfy (16*n^2-32*n+16+(36*n^3-92*n^2+48*n)*x+(9*n^4-18*n^3-33*n^2+42*n)*x^2-(9*n^3-4*n^2-7*n+2)*x^3+n^2*x^4)*P(n,x) = (16*n^2*x+(36*n^3+16*n^2-28*n-8)*x^2+(9*n^4+18*n^3-33*n^2-42*n)*x^3-(9*n^3+23*n^2+12*n)*x^4+(n^2+2*n+1)*x^5)*P(n-2,x) + (32*n^2-64*n+32+(72*n^3-96*n^2-96*n+96)*x+(18*n^4+36*n^3-194*n^2-52*n+160)*x^2+(9*n^4-18*n^3-97*n^2+26*n+80)*x^3-(9*n^3+12*n^2-12*n-12)*x^4+(n^2+2*n+1)*x^5)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 4*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3 + 4*x;\nP[n_, x_] := P[n, x] = Expand[Together[((16*n^2*x + (36*n^3 + 16*n^2 - 28*n - 8)*x^2 + (9*n^4 + 18*n^3 - 33*n^2 - 42*n)*x^3 - (9*n^3 + 23*n^2 + 12*n)*x^4 + (n^2 + 2*n + 1)*x^5)*P[n - 2, x] + (32*n^2 - 64*n + 32 + (72*n^3 - 96*n^2 - 96*n + 96)*x + (18*n^4 + 36*n^3 - 194*n^2 - 52*n + 160)*x^2 + (9*n^4 - 18*n^3 - 97*n^2 + 26*n + 80)*x^3 - (9*n^3 + 12*n^2 - 12*n - 12)*x^4 + (n^2 + 2*n + 1)*x^5)*P[n - 1, x])/(16*n^2 - 32*n + 16 + (36*n^3 - 92*n^2 + 48*n)*x + (9*n^4 - 18*n^3 - 33*n^2 + 42*n)*x^2 - (9*n^3 - 4*n^2 - 7*n + 2)*x^3 + n^2*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] = 3 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = ((16*n^2*x + (36*n^3 + 16*n^2 - 28*n - 8)*x^2 + (9*n^4 + 18*n^3 - 33*n^2 - 42*n)*x^3 - (9*n^3 + 23*n^2 + 12*n)*x^4 + (n^2 + 2*n + 1)*x^5)*v[i - 2] + (32*n^2 - 64*n + 32 + (72*n^3 - 96*n^2 - 96*n + 96)*x + (18*n^4 + 36*n^3 - 194*n^2 - 52*n + 160)*x^2 + (9*n^4 - 18*n^3 - 97*n^2 + 26*n + 80)*x^3 - (9*n^3 + 12*n^2 - 12*n - 12)*x^4 + (n^2 + 2*n + 1)*x^5)*v[i - 1])/(16*n^2 - 32*n + 16 + (36*n^3 - 92*n^2 + 48*n)*x + (9*n^4 - 18*n^3 - 33*n^2 + 42*n)*x^2 - (9*n^3 - 4*n^2 - 7*n + 2)*x^3 + n^2*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-d6b7687ba0ff59dd","steps":["(\u03b8+1)\u00b2"],"edges":["e-8755b73a3425b06c"],"text":"A116412 \u2192 (\u03b8+1)\u00b2"},"headline":"A116412 \u2192 (\u03b8+1)\u00b2","chain":null}