{"id":"n-b411a18788d1c9a6","revision":0,"first_index":1,"fingerprint":"f6-ecf0299b922a6878","name":"theta_plus_one of basis_map_A013611 of Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 2, -1, 0, 0, 0, 0, 0, ...] DELTA [1, 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":"affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["1","8"],["3","32","48"],["7","104","336","256"],["17","320","1536","2560","1280"],["41","936","5952","15360","16640","6144"],["99","2656","20976","73216","124160","98304","28672"],["239","7368","69456","305408","707840","878592","544768","131072"],["577","20096","220032","1166336","3432960","5849088","5677056","2883584","589824"],["1393","54088","674304","4182016","14932480","32452608","43294720","34340864","14745600","2621440"],["3363","144032","2013744","14300160","60039680","158908416","271007744","295436288","197591040","73400320","11534336"],["8119","380200","5890512","47118592","227422720","710135808","1478328320","2062024704","1893924864","1093140480","357564416","50331648"],["19601","996288","16940160","150693376","821978880","2958999552","7283204096","12416450560","14594015232","11555307520","5859442688","1711276032","218103808"],["47321","2594536","48029760","470268928","2860440320","11665262592","33167368192","66947645440","96235094016","97483489280","67729620992","30601641984","8069840896","939524096"],["114243","6720736","134544624","1437737472","9647671040","43965136896","141864497152","331228643328","564202635264","699096104960","621089390592","384030474240","156380430336","37580963840","4026531840"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","9","83","703","5713","45073","348083","2644743","19839553","147299513","1084379123","7926270383","57587427793","416220868353","2994691596563"],"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","8"],["3","32","48"],["7","104","336","256"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-1","-8","-16"]]},"deriv_order":0,"offset":4,"sign":"none"},{"coeff":{"coeffs":[["-4","-24","-32"]]},"deriv_order":0,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["-2","-8","-16"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["4","8"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":181406,"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-a137c7582d7f34ac"]},{"id":181407,"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-a137c7582d7f34ac"]},{"id":181408,"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-a137c7582d7f34ac"]}],"region_exact":{},"incoming_edges":[{"id":"e-a137c7582d7f34ac","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-b411a18788d1c9a6","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":"3efa0cf791f82475653c74abb323ebad6ae409d9831a44fa0da650eb1894b651","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-d7230c9910e765ce","role":"sequence","name":"basis_map_A013611 of Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 2, -1, 0, 0, 0, 0, 0, ...] DELTA [1, 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:    1,     8;\n   3:    3,    32,     48;\n   4:    7,   104,    336,     256;\n   5:   17,   320,   1536,    2560,     1280;\n   6:   41,   936,   5952,   15360,    16640,     6144;\n   7:   99,  2656,  20976,   73216,   124160,    98304,    28672;\n   8:  239,  7368,  69456,  305408,   707840,   878592,   544768,   131072;\n   9:  577, 20096, 220032, 1166336,  3432960,  5849088,  5677056,  2883584,   589824;\n  10: 1393, 54088, 674304, 4182016, 14932480, 32452608, 43294720, 34340864, 14745600, 2621440;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(1+8*x+16*x^2)*P(n-4,x) - (4+24*x+32*x^2)*P(n-3,x) - (2+8*x+16*x^2)*P(n-2,x) + (4+8*x)*P(n-1,x) for n >= 5, with P(1,x) = 1, P(2,x) = 1 + 8*x, P(3,x) = 3 + 32*x + 48*x^2, P(4,x) = 7 + 104*x + 336*x^2 + 256*x^3.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 1 + 8*x;\nP[3, x_] := 3 + 32*x + 48*x^2;\nP[4, x_] := 7 + 104*x + 336*x^2 + 256*x^3;\nP[n_, x_] := P[n, x] = Expand[-(1 + 8*x + 16*x^2)*P[n - 4, x] - (4 + 24*x + 32*x^2)*P[n - 3, x] - (2 + 8*x + 16*x^2)*P[n - 2, x] + (4 + 8*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + 8*x; v[3] = 3 + 32*x + 48*x^2; v[4] = 7 + 104*x + 336*x^2 + 256*x^3; for(n = 5, N, my(i = n); v[i] = -(1 + 8*x + 16*x^2)*v[i - 4] - (4 + 24*x + 32*x^2)*v[i - 3] - (2 + 8*x + 16*x^2)*v[i - 2] + (4 + 8*x)*v[i - 1]); 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-b0fc620949fb5990","steps":["basis A013611","\u03b8+1"],"edges":["e-458ebe1b0b79e88d","e-a137c7582d7f34ac"],"text":"A122542 \u2192 basis A013611 \u2192 \u03b8+1"},"headline":"A122542 \u2192 basis A013611 \u2192 \u03b8+1","chain":null}