{"id":"n-ea8bb83df35fc654","revision":0,"first_index":1,"fingerprint":"f6-d2fcce66244b0ee8","name":"theta_plus_one of Triangle T(n,k), read by rows, given by (2,1/2,3/2,0,0,0,0,0,0,0,...) DELTA (0,1/2,-1/2,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"half_affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["2"],["5","2"],["14","12"],["41","52","3"],["122","200","30"],["365","726","189","4"],["1094","2548","966","56"],["3281","8744","4386","464","5"],["9842","29520","18468","2976","90"],["29525","98410","73830","16360","925","6"],["88574","324764","284196","81136","7150","132"],["265721","1062876","1062909","373904","46275","1620","7"],["797162","3454360","3886194","1631136","265070","14664","182"],["2391485","11160254","13950363","6820044","1388835","109326","2597","8"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","2","7","26","96","352","1284","4664","16880","60896","219056","785952","2813312","10048768","35822912"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"],["2"]],"recurrence":{"denominator":{"coeffs":[["96","88","0"],["-48","-60","4"],["0","8","-6"],["0","0","2"]]},"terms":[{"coeff":{"coeffs":[["240","280","88","0"],["-120","-168","-50","-2"],["0","14","-7","-1"],["0","0","5","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-72","-180","-64","44","0"],["0","24","22","-4","-2"],["0","0","6","4","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":174724,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-cc2faf9506fd4f29"]},{"id":174725,"position":1,"claim":"one_signed_coefficients","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":1},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 1: the coefficients of P_n have one sign","region_exact":null,"edges":["e-cc2faf9506fd4f29"]},{"id":174726,"position":2,"claim":"hadamard_kernel","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":1},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-cc2faf9506fd4f29"]}],"region_exact":{},"incoming_edges":[{"id":"e-cc2faf9506fd4f29","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"euler_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-ea8bb83df35fc654","premise_count":1,"theorem":"Laguerre's theorem on multiplier sequences: if f is a real polynomial whose roots are all real and <= 0, then (f(k))_(k >= 0) is a multiplier sequence, so x^k -> f(k) x^k maps every real-rooted polynomial to a real-rooted polynomial or zero (Laguerre; see T. 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":"9dab2ea9d8fb4bd8b00c61f07faca557ce7d07685ea7fadf1878a7f29363a461","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-048e7c9e10af99dd","role":"sequence","name":"Triangle T(n,k), read by rows, given by (2,1/2,3/2,0,0,0,0,0,0,0,...) DELTA (0,1/2,-1/2,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938.","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;\n   3:    5,     2;\n   4:   14,    12;\n   5:   41,    52,     3;\n   6:  122,   200,    30;\n   7:  365,   726,   189,    4;\n   8: 1094,  2548,   966,   56;\n   9: 3281,  8744,  4386,  464,  5;\n  10: 9842, 29520, 18468, 2976, 90;\n","recurrence_sentence":"The row polynomials satisfy -(48*n-96-(8*n^2-60*n+88)*x-(2*n^3-6*n^2+4*n)*x^2)*P(n,x) = -(120*n-240-(14*n^2-168*n+280)*x-(5*n^3-7*n^2-50*n+88)*x^2-(n^3-n^2-2*n)*x^3)*P(n-1,x) - (72-(24*n-180)*x-(6*n^2+22*n-64)*x^2-(4*n^2-4*n+44)*x^3+(2*n^2+2*n)*x^4)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[Together[(-(120*n - 240 - (14*n^2 - 168*n + 280)*x - (5*n^3 - 7*n^2 - 50*n + 88)*x^2 - (n^3 - n^2 - 2*n)*x^3)*P[n - 1, x] - (72 - (24*n - 180)*x - (6*n^2 + 22*n - 64)*x^2 - (4*n^2 - 4*n + 44)*x^3 + (2*n^2 + 2*n)*x^4)*D[P[n - 1, x], x])/(-48*n + 96 + (8*n^2 - 60*n + 88)*x + (2*n^3 - 6*n^2 + 4*n)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2; for(n = 3, N, my(i = n); v[i] = (-(120*n - 240 - (14*n^2 - 168*n + 280)*x - (5*n^3 - 7*n^2 - 50*n + 88)*x^2 - (n^3 - n^2 - 2*n)*x^3)*v[i - 1] - (72 - (24*n - 180)*x - (6*n^2 + 22*n - 64)*x^2 - (4*n^2 - 4*n + 44)*x^3 + (2*n^2 + 2*n)*x^4)*deriv(v[i - 1], x))/(-48*n + 96 + (8*n^2 - 60*n + 88)*x + (2*n^3 - 6*n^2 + 4*n)*x^2)); 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-048e7c9e10af99dd","steps":["\u03b8+1"],"edges":["e-cc2faf9506fd4f29"],"text":"A201730 \u2192 \u03b8+1"},"headline":"A201730 \u2192 \u03b8+1","chain":null}