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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":"8388da2811ba601cf823e00eb13403555191128423990a047df01271cb2e2852","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-d366c5ae31cc5546","role":"sequence","name":"basis_map_A013611 of delannoy","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-9094a75c83a9508f","kind":"fingerprint_related","node_a_id":"n-77a81144c7cca672","node_b_id":"n-30895e5d03a9a6e2","fingerprint":"f6-89ae647762cb208b","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-89ae647762cb208b"},"other_id":"n-77a81144c7cca672","other":{"id":"n-77a81144c7cca672","name":"theta_plus_one_squared of basis_map_A013611 of Square array of Delannoy numbers D(i,j) (i >= 0, j >= 0) read by antidiagonals.","first_index":1,"fingerprint":"f6-89ae647762cb208b","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"],["12","288","1152","1024"]],"row_sums_preview":["1","18","229","2476","24381","225654","1998185","17115864"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-811c78acf96cbc19","steps":["basis A013611","(\u03b8+1)\u00b2"],"edges":["e-8e7f1943c60a0009","e-0c48348bb8b297a8"],"text":"A008288 \u2192 basis A013611 \u2192 (\u03b8+1)\u00b2"},"headline":"A008288 \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:   12,    288,    1152,     1024;\n  4:   29,    928,    5760,    11264,     6400;\n  5:   70,   2800,   23616,    72704,    89600,     36864;\n  6:  169,   8112,   86544,   365568,   710400,    626688,    200704;\n  7:  408,  22848,  294912,  1587200,  4249600,   5898240,   4014080,   1048576;\n  8:  985,  63040,  955008,  6252544, 21414400,  40992768,  43753472,  24117248,   5308416;\n  9: 2378, 171216, 2977920, 22980608, 96076800, 235708416, 346816512, 298844160, 138018816, 26214400;\n","recurrence_sentence":"The row polynomials satisfy (1+(12*n-2)*x+(44*n^2-4*n-8)*x^2+(48*n^3+40*n^2-56*n-4)*x^3+(16*n^4+32*n^3-64*n^2-32*n+16)*x^4-(32*n^3+32*n^2)*x^5)*P(n,x) = (1+(12*n+14)*x+(44*n^2+132*n+72)*x^2+(48*n^3+360*n^2+504*n+156)*x^3+(16*n^4+288*n^3+864*n^2+672*n+80)*x^4+(64*n^4+352*n^3+384*n^2-160*n-192)*x^5-(128*n^3+512*n^2+640*n+256)*x^6)*P(n-2,x) + (2+(24*n+12)*x+(88*n^2+128*n+8)*x^2+(96*n^3+400*n^2+192*n-120)*x^3+(32*n^4+320*n^3+480*n^2-160*n-288)*x^4+(64*n^4+192*n^3-160*n^2-512*n-224)*x^5-(128*n^3+384*n^2+384*n+128)*x^6)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 16*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + 16*x;\nP[n_, x_] := P[n, x] = Expand[Together[((1 + (12*n + 14)*x + (44*n^2 + 132*n + 72)*x^2 + (48*n^3 + 360*n^2 + 504*n + 156)*x^3 + (16*n^4 + 288*n^3 + 864*n^2 + 672*n + 80)*x^4 + (64*n^4 + 352*n^3 + 384*n^2 - 160*n - 192)*x^5 - (128*n^3 + 512*n^2 + 640*n + 256)*x^6)*P[n - 2, x] + (2 + (24*n + 12)*x + (88*n^2 + 128*n + 8)*x^2 + (96*n^3 + 400*n^2 + 192*n - 120)*x^3 + (32*n^4 + 320*n^3 + 480*n^2 - 160*n - 288)*x^4 + (64*n^4 + 192*n^3 - 160*n^2 - 512*n - 224)*x^5 - (128*n^3 + 384*n^2 + 384*n + 128)*x^6)*P[n - 1, x])/(1 + (12*n - 2)*x + (44*n^2 - 4*n - 8)*x^2 + (48*n^3 + 40*n^2 - 56*n - 4)*x^3 + (16*n^4 + 32*n^3 - 64*n^2 - 32*n + 16)*x^4 - (32*n^3 + 32*n^2)*x^5)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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