{"id":"n-e9807d8de7393cf6","revision":0,"first_index":1,"fingerprint":"f6-bc9ef0e3331cfb9e","name":"theta_plus_one of Triangle read by rows, S_3(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >= 0.","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"],["2","2"],["4","14","3"],["8","78","45","4"],["16","406","477","104","5"],["32","2062","4425","1780","200","6"],["64","10374","38493","25480","5025","342","7"],["128","51998","323505","331604","102200","11844","539","8"],["256","260246","2666037","4078984","1845305","323988","24598","800","9"],["512","1301742","21718665","48423540","30933000","7722162","869946","46512","1134","10"],["1024","6509734","175701933","561617160","493591425","168396354","26408109","2064000","81810","1550","11"],["2048","32550718","1415380065","6412058004","7612301400","3455047728","724624593","77652696","4449060","135850","2057","12"],["4096","162757686","11371866597","72419811464","114587292105","67870573056","18523380876","2614154400","203034843","8883050","215259","2664","13"],["8192","813796622","91219069305","811780414900","1694746853800","1291304492478","449649952752","81295129344","8219829618","483202720","16659643","328068","3380","14"],["16384","4068999494","730973249373","9051209989640","24741181471825","23985872596686","10499520962931","2383673635200","305172590580","23146022900","1064631568","29656536","483847","4214","15"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","4","21","135","1008","8505","79785","821826","9200223","111017223","1434373110","19734204231","287761976109","4429529740836","71725914311193"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"],["2","2"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["2","2","-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","3","-3"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["1","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","3"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":180731,"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-d018fd65aa087e62"]},{"id":180732,"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-d018fd65aa087e62"]},{"id":180733,"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-d018fd65aa087e62"]},{"id":180734,"position":3,"claim":"interlacing","from_index":2,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2","region_exact":null,"edges":["e-d018fd65aa087e62"]}],"region_exact":{},"incoming_edges":[{"id":"e-d018fd65aa087e62","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-e9807d8de7393cf6","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":"99e79a6ff444431b33cc55408f38c0dcb62a148a9d80ba3b9ebbfb07b41abab4","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-58f68e861f34b78a","role":"sequence","name":"Triangle read by rows, S_3(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >= 0.","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:   2,       2;\n   3:   4,      14,        3;\n   4:   8,      78,       45,        4;\n   5:  16,     406,      477,      104,        5;\n   6:  32,    2062,     4425,     1780,      200,       6;\n   7:  64,   10374,    38493,    25480,     5025,     342,      7;\n   8: 128,   51998,   323505,   331604,   102200,   11844,    539,     8;\n   9: 256,  260246,  2666037,  4078984,  1845305,  323988,  24598,   800,    9;\n  10: 512, 1301742, 21718665, 48423540, 30933000, 7722162, 869946, 46512, 1134, 10;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2+2*x-x^2)*P(n-2,x) + (3*x-3*x^2)*d/dx P(n-2,x) + (1+2*x)*P(n-1,x) + 3*x*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2 + 2*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 2 + 2*x;\nP[n_, x_] := P[n, x] = Expand[(2 + 2*x - x^2)*P[n - 2, x] + (3*x - 3*x^2)*D[P[n - 2, x], x] + (1 + 2*x)*P[n - 1, x] + 3*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] = 2 + 2*x; for(n = 3, N, my(i = n); v[i] = (2 + 2*x - x^2)*v[i - 2] + (3*x - 3*x^2)*deriv(v[i - 2], x) + (1 + 2*x)*v[i - 1] + 3*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-58f68e861f34b78a","steps":["\u03b8+1"],"edges":["e-d018fd65aa087e62"],"text":"A225468 \u2192 \u03b8+1"},"headline":"A225468 \u2192 \u03b8+1","chain":null}
