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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":"0443b75e152f0463618063356b75f5cfba48c6e890eaed3bceda3027e0a055e0","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-333a80cbe99b35f1","role":"sequence","name":"basis_map_A013611 of Triangle of Narayana numbers T(n,k) = C(n-1,k-1)*C(n,k-1)/k with 1 <= k <= n, read by rows. Also called the Catalan triangle.","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":[{"id":"l-bca6b9c2ada49986","kind":"fingerprint_related","node_a_id":"n-b991dcbedf3c1d83","node_b_id":"n-035bf07f85d18538","fingerprint":"f6-777088b2ee260e77","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-777088b2ee260e77"},"other_id":"n-035bf07f85d18538","other":{"id":"n-035bf07f85d18538","name":"theta_plus_one_squared of basis_map_A013611 of narayana","first_index":1,"fingerprint":"f6-777088b2ee260e77","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","16"],["5","80","144"]],"card_rows":[["1"],["2","16"],["5","80","144"],["14","336","1296","1024"]],"row_sums_preview":["1","18","229","2670","30186","336356","3715693","40801494"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-2a05d8340c329b21","steps":["basis A013611","(\u03b8+1)\u00b2"],"edges":["e-98249f41ee5a34be","e-d02f6c8d2b8a9699"],"text":"narayana \u2192 basis A013611 \u2192 (\u03b8+1)\u00b2"},"headline":"narayana \u2192 basis A013611 \u2192 (\u03b8+1)\u00b2","chain":null}}],"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:     2,      16;\n  2:     5,      80,      144;\n  3:    14,     336,     1296,      1024;\n  4:    42,    1344,     8064,     14336,      6400;\n  5:   132,    5280,    43200,    122880,    128000,      36864;\n  6:   429,   20592,   213840,    844800,   1440000,     995328,     200704;\n  7:  1430,   80080,  1009008,   5125120,  12320000,   14192640,    7024640,   1048576;\n  8:  4862,  311168,  4612608,  28700672,  89689600,  147603456,  123633664,  46137344,   5308416;\n  9: 16796, 1209312, 20621952, 152076288, 587059200, 1268047872, 1534181376, 981467136, 286654464, 26214400;\n","recurrence_sentence":"The row polynomials satisfy (n^3+4*n^2+5*n+2+(6*n^4+21*n^3+19*n^2-4)*x+(4*n^5+16*n^4+20*n^3+8*n^2)*x^2)*P(n,x) = (4*n^3+10*n^2+8*n+2+(24*n^4+96*n^3+118*n^2+42*n-4)*x+(16*n^5+160*n^4+360*n^3+232*n^2)*x^2+(48*n^5+224*n^4+320*n^3+144*n^2+32*n)*x^3)*P(n-1,x) - ((2*n^2+4*n+2)*x+(12*n^3+44*n^2+47*n+14)*x^2+(8*n^4+88*n^3+208*n^2+160*n+32)*x^3+(32*n^4+160*n^3+256*n^2+144*n+32)*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^3 + 10*n^2 + 8*n + 2 + (24*n^4 + 96*n^3 + 118*n^2 + 42*n - 4)*x + (16*n^5 + 160*n^4 + 360*n^3 + 232*n^2)*x^2 + (48*n^5 + 224*n^4 + 320*n^3 + 144*n^2 + 32*n)*x^3)*P[n - 1, x] - ((2*n^2 + 4*n + 2)*x + (12*n^3 + 44*n^2 + 47*n + 14)*x^2 + (8*n^4 + 88*n^3 + 208*n^2 + 160*n + 32)*x^3 + (32*n^4 + 160*n^3 + 256*n^2 + 144*n + 32)*x^4)*D[P[n - 1, x], x])/(n^3 + 4*n^2 + 5*n + 2 + (6*n^4 + 21*n^3 + 19*n^2 - 4)*x + (4*n^5 + 16*n^4 + 20*n^3 + 8*n^2)*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^3 + 10*n^2 + 8*n + 2 + (24*n^4 + 96*n^3 + 118*n^2 + 42*n - 4)*x + (16*n^5 + 160*n^4 + 360*n^3 + 232*n^2)*x^2 + (48*n^5 + 224*n^4 + 320*n^3 + 144*n^2 + 32*n)*x^3)*v[i - 1] - ((2*n^2 + 4*n + 2)*x + (12*n^3 + 44*n^2 + 47*n + 14)*x^2 + (8*n^4 + 88*n^3 + 208*n^2 + 160*n + 32)*x^3 + (32*n^4 + 160*n^3 + 256*n^2 + 144*n + 32)*x^4)*deriv(v[i - 1], x))/(n^3 + 4*n^2 + 5*n + 2 + (6*n^4 + 21*n^3 + 19*n^2 - 4)*x + (4*n^5 + 16*n^4 + 20*n^3 + 8*n^2)*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-c2eaf5212a4dec13","steps":["basis A013611","(\u03b8+1)\u00b2"],"edges":["e-c1ad06aed8ab4413","e-198a9f0995d8a3e0"],"text":"A001263 \u2192 basis A013611 \u2192 (\u03b8+1)\u00b2"},"headline":"A001263 \u2192 basis A013611 \u2192 (\u03b8+1)\u00b2","chain":null}