{"id":"n-88ca4c00ef12fd06","revision":0,"first_index":0,"fingerprint":"f6-0e03fcf0cb29044a","name":"theta_plus_one of veronese_section_2_0 of Triangle whose (i,j)-th entry is binomial(i,j)*2^(i-j).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"half_affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["2"],["4","2"],["8","12"],["16","48","3"],["32","160","30"],["64","480","180","4"],["128","1344","840","56"],["256","3584","3360","448","5"],["512","9216","12096","2688","90"],["1024","23040","40320","13440","900","6"],["2048","56320","126720","59136","6600","132"],["4096","135168","380160","236544","39600","1584","7"],["8192","319488","1098240","878592","205920","13728","182"],["16384","745472","3075072","3075072","960960","96096","2548","8"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","2","6","20","67","222","728","2368","7653","24602","78730","250956","797159","2524342","7971612"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["2"]],"recurrence":{"denominator":{"coeffs":[["48","0"],["-48","-2"],["0","0"],["0","2"]]},"terms":[{"coeff":{"coeffs":[["96","48","-2"],["-96","-52","-1"],["0","0","2"],["0","4","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-192","64","-4"],["0","0","24","-6"],["0","0","8","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":173855,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-a1fa35951cbade22"]},{"id":173856,"position":1,"claim":"one_signed_coefficients","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":0},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 0: the coefficients of P_n have one sign","region_exact":null,"edges":["e-a1fa35951cbade22"]},{"id":173857,"position":2,"claim":"hadamard_kernel","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":0},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-a1fa35951cbade22"]}],"region_exact":{},"incoming_edges":[{"id":"e-a1fa35951cbade22","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-88ca4c00ef12fd06","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":"a7d937b4ccaa107f405bc6488f39c898306bf3ea8374137de86bd4256ff5f8cc","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-b6912a345c4fb865","role":"sequence","name":"veronese_section_2_0 of Triangle whose (i,j)-th entry is binomial(i,j)*2^(i-j).","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-d1199ac05fad2936","kind":"fingerprint_related","node_a_id":"n-88ca4c00ef12fd06","node_b_id":"n-8171ffa220e5c9a0","fingerprint":"f6-0e03fcf0cb29044a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-0e03fcf0cb29044a"},"other_id":"n-8171ffa220e5c9a0","other":{"id":"n-8171ffa220e5c9a0","name":"theta_plus_one of veronese_section_2_0 of Triangle read by rows: T(n,k) is the number of ternary words of length n having k levels (n >= 1, 0 <= k <= n-1). A level is a pair of identical consecutive letters.","first_index":0,"fingerprint":"f6-0e03fcf0cb29044a","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["3"],["6"],["12","6"]],"card_rows":[["3"],["6"],["12","6"],["24","36"]],"row_sums_preview":["3","6","18","60","201","666","2184","7104"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-51b4f5e7901c2b50","steps":["Veronese 2","\u03b8+1"],"edges":["e-74cdb64b6d1e6b98","e-c2ff94203ced9e90"],"text":"A120910 \u2192 Veronese 2 \u2192 \u03b8+1"},"headline":"A120910 \u2192 Veronese 2 \u2192 \u03b8+1","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;\n  2:   4,    2;\n  3:   8,   12;\n  4:  16,   48,     3;\n  5:  32,  160,    30;\n  6:  64,  480,   180,    4;\n  7: 128, 1344,   840,   56;\n  8: 256, 3584,  3360,  448,  5;\n  9: 512, 9216, 12096, 2688, 90;\n","recurrence_sentence":"The row polynomials satisfy -(48*n-48-(2*n^3-2*n)*x)*P(n,x) = -(96*n-96-(4*n^3-52*n+48)*x-(n^3+2*n^2-n-2)*x^2)*P(n-1,x) - (192*x-(8*n^2+24*n+64)*x^2+(2*n^2+6*n+4)*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[Together[(-(96*n - 96 - (4*n^3 - 52*n + 48)*x - (n^3 + 2*n^2 - n - 2)*x^2)*P[n - 1, x] - (192*x - (8*n^2 + 24*n + 64)*x^2 + (2*n^2 + 6*n + 4)*x^3)*D[P[n - 1, x], x])/(-48*n + 48 + (2*n^3 - 2*n)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = (-(96*n - 96 - (4*n^3 - 52*n + 48)*x - (n^3 + 2*n^2 - n - 2)*x^2)*v[i - 1] - (192*x - (8*n^2 + 24*n + 64)*x^2 + (2*n^2 + 6*n + 4)*x^3)*deriv(v[i - 1], x))/(-48*n + 48 + (2*n^3 - 2*n)*x)); 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-b6912a345c4fb865","steps":["\u03b8+1"],"edges":["e-a1fa35951cbade22"],"text":"A372868 \u2192 \u03b8+1"},"headline":"A372868 \u2192 \u03b8+1","chain":null}