{"id":"n-b2d37c586fa5ee60","revision":0,"first_index":0,"fingerprint":"f6-b56b614d84ca9e33","name":"theta_plus_one of Triangle of coefficients in expansion of (x+1)*(x+3)*...*(x + 2n - 1) in rising powers of x.","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"],["1","2"],["3","8","3"],["15","46","27","4"],["105","352","258","64","5"],["945","3378","2850","920","125","6"],["10395","39048","36417","13920","2525","216","7"],["135135","528414","531993","229516","50225","5838","343","8"],["2027025","8196480","8772516","4152064","1040270","147840","11956","512","9"],["34459425","143394210","161427492","82281776","22874670","3761604","375732","22368","729","10"],["654729075","2793408840","3282213663","1778590400","537470950","98920080","11527446","854400","39015","1000","11"],["13749310575","59971043790","73116600183","41726683284","13510127950","2722286820","357483126","31116624","1780515","64350","1331","12"],["316234143225","1406832628320","1771638369894","1057202515776","362891296955","78824750400","11398113188","1124234496","75958047","3458400","101398","1728","13"],["7905853580625","35803283994450","46401208189830","28792247387592","10393785568595","2406088316346","376915038500","41132277472","3163714983","170857830","6339190","153816","2197","14"],["213458046676875","982500375011400","1306537547117085","839258957051424","316622519586555","77436927223656","12983809075237","1541331535744","131694116697","8128400280","359101743","11068512","225953","2744","15"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","3","14","92","784","8224","102528","1481472","24348672","448598016","9157754880","205186498560","5006225571840","132123855421440","3750479706193920"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["1","2"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-9","6","-1"],["12","-4","0"],["-4","0","0"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-4","2"],["4","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":182719,"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-972afaae00abf9d3"]},{"id":182720,"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-972afaae00abf9d3"]},{"id":182721,"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-972afaae00abf9d3"]},{"id":182722,"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-972afaae00abf9d3"]}],"region_exact":{},"incoming_edges":[{"id":"e-972afaae00abf9d3","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-b2d37c586fa5ee60","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":"f6ff8a68fcdd2700b6a305980839c450780ff9fd5d133fb19dccb27597533a97","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-4f6e747b794bcc2e","role":"sequence","name":"Triangle of coefficients in expansion of (x+1)*(x+3)*...*(x + 2n - 1) in rising powers of x.","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-f10c79de63a5984d","kind":"fingerprint_related","node_a_id":"n-b2d37c586fa5ee60","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-79285e9af5d117ab","other":{"id":"n-79285e9af5d117ab","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.","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"],["2","4"],["12","32","12"]],"card_rows":[["1"],["2","4"],["12","32","12"],["120","368","216","32"]],"row_sums_preview":["1","6","56","736","12544","263168","6561792","189628416"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"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}}],"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:        1,         2;\n  2:        3,         8,         3;\n  3:       15,        46,        27,        4;\n  4:      105,       352,       258,       64,        5;\n  5:      945,      3378,      2850,      920,      125,       6;\n  6:    10395,     39048,     36417,    13920,     2525,     216,      7;\n  7:   135135,    528414,    531993,   229516,    50225,    5838,    343,     8;\n  8:  2027025,   8196480,   8772516,  4152064,  1040270,  147840,  11956,   512,   9;\n  9: 34459425, 143394210, 161427492, 82281776, 22874670, 3761604, 375732, 22368, 729, 10;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(4*n^2-12*n+9+(4*n-6)*x+x^2)*P(n-2,x) + (4*n-4+2*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 2*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 2*x;\nP[n_, x_] := P[n, x] = Expand[-(4*n^2 - 12*n + 9 + (4*n - 6)*x + x^2)*P[n - 2, x] + (4*n - 4 + 2*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] = 1 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = -(4*n^2 - 12*n + 9 + (4*n - 6)*x + x^2)*v[i - 2] + (4*n - 4 + 2*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-4f6e747b794bcc2e","steps":["\u03b8+1"],"edges":["e-972afaae00abf9d3"],"text":"A028338 \u2192 \u03b8+1"},"headline":"A028338 \u2192 \u03b8+1","chain":null}