{"id":"n-254cbc520517186c","revision":0,"first_index":1,"fingerprint":"f6-090aca1ee1bd5ff6","name":"theta_plus_one of veronese_section_2_0 of Triangle read by rows: T(n, k) = binomial(2n, 2k).","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"],["1"],["1","2"],["1","30"],["1","140","3"],["1","420","135"],["1","990","1485","4"],["1","2002","9009","364"],["1","3640","38610","7280","5"],["1","6120","131274","74256","765"],["1","9690","377910","503880","24225","6"],["1","14630","959310","2586584","373065","1386"],["1","21252","2206413","10816624","3677355","63756","7"],["1","29900","4686825","38630800","26558675","1381380","2275"],["1","40950","9324315","121687020","152108775","18648630","143325","8"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","1","3","31","144","556","2480","11376","49536","212416","915712","3934976","16785408","71289856","301953024"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"],["1"],["1","2"],["1","30"],["1","140","3"],["1","420","135"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["191","-381","189","1"]]},"deriv_order":0,"offset":6,"sign":"none"},{"coeff":{"coeffs":[["-64","192","-192","64"]]},"deriv_order":1,"offset":6,"sign":"none"},{"coeff":{"coeffs":[["3342","-2252","-66"]]},"deriv_order":0,"offset":5,"sign":"none"},{"coeff":{"coeffs":[["-128","1264","-1120","-16"]]},"deriv_order":1,"offset":5,"sign":"none"},{"coeff":{"coeffs":[["9041","146","-3"]]},"deriv_order":0,"offset":4,"sign":"none"},{"coeff":{"coeffs":[["1152","5184","64"]]},"deriv_order":1,"offset":4,"sign":"none"},{"coeff":{"coeffs":[["1860","52"]]},"deriv_order":0,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["-2048","416","32"]]},"deriv_order":1,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["-111","3"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["1472","128"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["14"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-384","-16"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":175836,"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-6e92ea2273380d80"]},{"id":175837,"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-6e92ea2273380d80"]},{"id":175838,"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-6e92ea2273380d80"]}],"region_exact":{},"incoming_edges":[{"id":"e-6e92ea2273380d80","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-254cbc520517186c","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":"b08e47cd7c03ba6fde95799ed80e91fa89ac45cca5ad9c9d8b42ddeff5f72379","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-948c4fae35fcbf15","role":"sequence","name":"veronese_section_2_0 of Triangle read by rows: T(n, k) = binomial(2n, 2k).","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: 1;\n   3: 1,    2;\n   4: 1,   30;\n   5: 1,  140,      3;\n   6: 1,  420,    135;\n   7: 1,  990,   1485,     4;\n   8: 1, 2002,   9009,   364;\n   9: 1, 3640,  38610,  7280,   5;\n  10: 1, 6120, 131274, 74256, 765;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (191-381*x+189*x^2+x^3)*P(n-6,x) - (64-192*x+192*x^2-64*x^3)*d/dx P(n-6,x) + (3342-2252*x-66*x^2)*P(n-5,x) - (128-1264*x+1120*x^2+16*x^3)*d/dx P(n-5,x) + (9041+146*x-3*x^2)*P(n-4,x) + (1152+5184*x+64*x^2)*d/dx P(n-4,x) + (1860+52*x)*P(n-3,x) - (2048-416*x-32*x^2)*d/dx P(n-3,x) - (111-3*x)*P(n-2,x) + (1472+128*x)*d/dx P(n-2,x) + 14*P(n-1,x) - (384+16*x)*d/dx P(n-1,x) for n >= 7, with P(1,x) = 1, P(2,x) = 1, P(3,x) = 1 + 2*x, P(4,x) = 1 + 30*x, P(5,x) = 1 + 140*x + 3*x^2, P(6,x) = 1 + 420*x + 135*x^2.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 1;\nP[3, x_] := 1 + 2*x;\nP[4, x_] := 1 + 30*x;\nP[5, x_] := 1 + 140*x + 3*x^2;\nP[6, x_] := 1 + 420*x + 135*x^2;\nP[n_, x_] := P[n, x] = Expand[(191 - 381*x + 189*x^2 + x^3)*P[n - 6, x] - (64 - 192*x + 192*x^2 - 64*x^3)*D[P[n - 6, x], x] + (3342 - 2252*x - 66*x^2)*P[n - 5, x] - (128 - 1264*x + 1120*x^2 + 16*x^3)*D[P[n - 5, x], x] + (9041 + 146*x - 3*x^2)*P[n - 4, x] + (1152 + 5184*x + 64*x^2)*D[P[n - 4, x], x] + (1860 + 52*x)*P[n - 3, x] - (2048 - 416*x - 32*x^2)*D[P[n - 3, x], x] - (111 - 3*x)*P[n - 2, x] + (1472 + 128*x)*D[P[n - 2, x], x] + 14*P[n - 1, x] - (384 + 16*x)*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1; v[3] = 1 + 2*x; v[4] = 1 + 30*x; v[5] = 1 + 140*x + 3*x^2; v[6] = 1 + 420*x + 135*x^2; for(n = 7, N, my(i = n); v[i] = (191 - 381*x + 189*x^2 + x^3)*v[i - 6] - (64 - 192*x + 192*x^2 - 64*x^3)*deriv(v[i - 6], x) + (3342 - 2252*x - 66*x^2)*v[i - 5] - (128 - 1264*x + 1120*x^2 + 16*x^3)*deriv(v[i - 5], x) + (9041 + 146*x - 3*x^2)*v[i - 4] + (1152 + 5184*x + 64*x^2)*deriv(v[i - 4], x) + (1860 + 52*x)*v[i - 3] - (2048 - 416*x - 32*x^2)*deriv(v[i - 3], x) - (111 - 3*x)*v[i - 2] + (1472 + 128*x)*deriv(v[i - 2], x) + 14*v[i - 1] - (384 + 16*x)*deriv(v[i - 1], x)); 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-f7935c36a3c0ec8c","steps":["Veronese 2","\u03b8+1"],"edges":["e-d4161fce9e8d4a19","e-6e92ea2273380d80"],"text":"A086645 \u2192 Veronese 2 \u2192 \u03b8+1"},"headline":"A086645 \u2192 Veronese 2 \u2192 \u03b8+1","chain":null}