{"id":"n-79285e9af5d117ab","revision":0,"first_index":0,"fingerprint":"f6-b56b614d84ca9e33","name":"theta_plus_one of basis_map_A013609 of Triangle read by rows: T(n, k) is the Sheffer triangle ((1 - 4*x)^(-1/4), (-1/4)*log(1 - 4*x)). A generalized Stirling1 triangle.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["2","4"],["12","32","12"],["120","368","216","32"],["1680","5632","4128","1024","80"],["30240","108096","91200","29440","4000","192"],["665280","2499072","2330688","890880","161600","13824","448"],["17297280","67636992","68095104","29378048","6428800","747264","43904","1024"],["518918400","2098298880","2245764096","1062928384","266309120","37847040","3060736","131072","2304"],["17643225600","73417835520","82650875904","42128269312","11711831040","1925941248","192374784","11452416","373248","5120"],["670442572800","2860450652160","3360986790912","1821276569600","550370252800","101294161920","11804104704","874905600","39951360","1024000","11264"],["28158588057600","122820697681920","149742797174784","85456247365632","27668742041600","5575243407360","732125442048","63726845952","3646494720","131788800","2725888","24576"],["1295295050649600","5762386445598720","7256630763085824","4330301504618496","1486402752327680","322866177638400","46686671618048","4604864495616","311124160512","14165606400","415326208","7077888","53248"],["64764752532480000","293300502482534400","380118697491087360","235866090599153664","85145891377930240","19710675487506432","3087687995392000","336955617050624","25917153140736","1399667343360","51930644480","1260060672","17997824","114688"],["3497296636753920000","16097286144186777600","21406311171966320640","13750418752330530816","5187543360906117120","1268726615632379904","212726727888683008","25253175881629696","2157676407963648","133175710187520","5883522957312","181346500608","3702013952","44957696","245760"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","6","56","736","12544","263168","6561792","189628416","6233260032","229682184192","9377540997120","420221949050880","20505499942256640","1082358623612436480","61447859506281185280"],"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","4"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-36","24","-4"],["48","-16","0"],["-16","0","0"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-8","4"],["8","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":188692,"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-c27c083fcf4df19c"]},{"id":188693,"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-c27c083fcf4df19c"]},{"id":188694,"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-c27c083fcf4df19c"]},{"id":188695,"position":3,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1","region_exact":null,"edges":["e-c27c083fcf4df19c"]}],"region_exact":{},"incoming_edges":[{"id":"e-c27c083fcf4df19c","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-79285e9af5d117ab","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":"77c1945d61cfa29c037c3f3ce36124922885692e531002b62227d79e2c7e3330","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-de50e58c7c589630","role":"sequence","name":"basis_map_A013609 of Triangle read by rows: T(n, k) is the Sheffer triangle ((1 - 4*x)^(-1/4), (-1/4)*log(1 - 4*x)). A generalized Stirling1 triangle.","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-78af8245446e4966","kind":"fingerprint_related","node_a_id":"n-79285e9af5d117ab","node_b_id":"n-ff1221cd172491ae","fingerprint":"f6-b56b614d84ca9e33","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b56b614d84ca9e33"},"other_id":"n-ff1221cd172491ae","other":{"id":"n-ff1221cd172491ae","name":"theta_plus_one of basis_map_A007318 of Triangle T(n,k) giving number of degree-2n permutations which decompose into exactly k cycles of even length, k=0..n.","first_index":0,"fingerprint":"f6-b56b614d84ca9e33","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","2"],["9","24","9"]],"card_rows":[["1"],["1","2"],["9","24","9"],["225","690","405","60"]],"row_sums_preview":["1","3","42","1380","82320","7771680","1065778560","200198718720"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-1336d058dca9979e","steps":["basis A007318","\u03b8+1"],"edges":["e-857dad5fd144a15f","e-9f4e9fdc4b1f04d0"],"text":"A204420 \u2192 basis A007318 \u2192 \u03b8+1"},"headline":"A204420 \u2192 basis A007318 \u2192 \u03b8+1","chain":null}},{"id":"l-f832d8f8ae05e66d","kind":"fingerprint_related","node_a_id":"n-b2d37c586fa5ee60","node_b_id":"n-79285e9af5d117ab","fingerprint":"f6-b56b614d84ca9e33","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b56b614d84ca9e33"},"other_id":"n-b2d37c586fa5ee60","other":{"id":"n-b2d37c586fa5ee60","name":"theta_plus_one of Triangle of coefficients in expansion of (x+1)*(x+3)*...*(x + 2n - 1) in rising powers of x.","first_index":0,"fingerprint":"f6-b56b614d84ca9e33","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","2"],["3","8","3"]],"card_rows":[["1"],["1","2"],["3","8","3"],["15","46","27","4"]],"row_sums_preview":["1","3","14","92","784","8224","102528","1481472"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-4f6e747b794bcc2e","steps":["\u03b8+1"],"edges":["e-972afaae00abf9d3"],"text":"A028338 \u2192 \u03b8+1"},"headline":"A028338 \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,           4;\n  2:          12,          32,          12;\n  3:         120,         368,         216,          32;\n  4:        1680,        5632,        4128,        1024,          80;\n  5:       30240,      108096,       91200,       29440,        4000,        192;\n  6:      665280,     2499072,     2330688,      890880,      161600,      13824,       448;\n  7:    17297280,    67636992,    68095104,    29378048,     6428800,     747264,     43904,     1024;\n  8:   518918400,  2098298880,  2245764096,  1062928384,   266309120,   37847040,   3060736,   131072,   2304;\n  9: 17643225600, 73417835520, 82650875904, 42128269312, 11711831040, 1925941248, 192374784, 11452416, 373248, 5120;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(16*n^2-48*n+36+(16*n-24)*x+4*x^2)*P(n-2,x) + (8*n-8+4*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 4*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + 4*x;\nP[n_, x_] := P[n, x] = Expand[-(16*n^2 - 48*n + 36 + (16*n - 24)*x + 4*x^2)*P[n - 2, x] + (8*n - 8 + 4*x)*P[n - 1, 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 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = -(16*n^2 - 48*n + 36 + (16*n - 24)*x + 4*x^2)*v[i - 2] + (8*n - 8 + 4*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 - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-4172472abe953cea","steps":["basis A013609","\u03b8+1"],"edges":["e-294a6ed01a588998","e-c27c083fcf4df19c"],"text":"A290319 \u2192 basis A013609 \u2192 \u03b8+1"},"headline":"A290319 \u2192 basis A013609 \u2192 \u03b8+1","chain":null}
