{"id":"n-00ac3a99d38c717d","revision":1,"first_index":0,"fingerprint":"f6-e9f0de137c11a043","name":"theta_plus_one of Triangle T(n, k) is coefficient of z^n*w^k in 1/(1 - 2*z - 2*w - 2*z*w) read by rows in order 00, 10, 01, 20, 11, 02, ...","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"],["4","20","12"],["8","64","96","32"],["16","176","444","352","80"],["32","448","1608","2144","1120","192"],["64","1088","5088","9760","8480","3264","448"],["128","2560","14784","37376","46720","29568","8960","1024"],["256","5888","40512","127744","211280","191616","94528","23552","2304"],["512","13312","106368","402944","835360","1002432","705152","283648","59904","5120"],["1024","29696","270336","1197568","2997440","4516800","4196416","2395136","811008","148480","11264"],["2048","65536","669696","3399680","9996160","18232320","21271040","15993856","7649280","2232320","360448","24576"],["4096","143360","1625088","9306112","31485440","67649280","95623360","90199040","56673792","23265280","5958656","860160","53248"],["8192","311296","3876864","24731648","94735360","234854400","391637120","447585280","352281600","189470720","68012032","15507456","2023424","114688"],["16384","671744","9117696","64135168","274565120","772638720","1489116160","2008908800","1914577920","1287731200","604043264","192405504","39510016","4702208","245760"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","6","36","200","1068","5544","28192","141120","697680","3414752","16575168","79896960","382846912","1825150080","8662385664"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["0","0","0"],["-2","4","4"],["0","-3","3"]]},"terms":[{"coeff":{"coeffs":[["0","16","28","16"],["-4","-16","2","12"],["0","-6","-12","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","4","24","36","8"],["0","0","6","24","6"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":165213,"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-6cc6a3c9200dfa14"]},{"id":165214,"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-6cc6a3c9200dfa14"]},{"id":165215,"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-6cc6a3c9200dfa14"]},{"id":165216,"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-6cc6a3c9200dfa14"]}],"region_exact":{},"incoming_edges":[{"id":"e-6cc6a3c9200dfa14","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-00ac3a99d38c717d","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":"aa3da6b8272aa0edf4626dd05c6aab3dd9d986f4ce52cc4e3ca1943bf5836191","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-1bffaa6198344f93","role":"sequence","name":"Triangle T(n, k) is coefficient of z^n*w^k in 1/(1 - 2*z - 2*w - 2*z*w) read by rows in order 00, 10, 01, 20, 11, 02, ...","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":[],"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:   4,    20,     12;\n  3:   8,    64,     96,     32;\n  4:  16,   176,    444,    352,     80;\n  5:  32,   448,   1608,   2144,   1120,     192;\n  6:  64,  1088,   5088,   9760,   8480,    3264,    448;\n  7: 128,  2560,  14784,  37376,  46720,   29568,   8960,   1024;\n  8: 256,  5888,  40512, 127744, 211280,  191616,  94528,  23552,  2304;\n  9: 512, 13312, 106368, 402944, 835360, 1002432, 705152, 283648, 59904, 5120;\n","recurrence_sentence":"The row polynomials satisfy -(2*n+(3*n^2-4*n)*x-(3*n^2+4*n)*x^2)*P(n,x) = -(4*n+(6*n^2+16*n-16)*x+(12*n^2-2*n-28)*x^2-(12*n+16)*x^3)*P(n-1,x) + (4*x+(6*n+24)*x^2+(24*n+36)*x^3+(6*n+8)*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(4*n + (6*n^2 + 16*n - 16)*x + (12*n^2 - 2*n - 28)*x^2 - (12*n + 16)*x^3)*P[n - 1, x] + (4*x + (6*n + 24)*x^2 + (24*n + 36)*x^3 + (6*n + 8)*x^4)*D[P[n - 1, x], x])/(-2*n - (3*n^2 - 4*n)*x + (3*n^2 + 4*n)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (-(4*n + (6*n^2 + 16*n - 16)*x + (12*n^2 - 2*n - 28)*x^2 - (12*n + 16)*x^3)*v[i - 1] + (4*x + (6*n + 24)*x^2 + (24*n + 36)*x^3 + (6*n + 8)*x^4)*deriv(v[i - 1], x))/(-2*n - (3*n^2 - 4*n)*x + (3*n^2 + 4*n)*x^2)); 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-1bffaa6198344f93","steps":["\u03b8+1"],"edges":["e-6cc6a3c9200dfa14"],"text":"A059473 \u2192 \u03b8+1"},"headline":"A059473 \u2192 \u03b8+1","chain":null}