{"id":"n-b6060bc044d672a9","revision":0,"first_index":0,"fingerprint":"f6-360a628193aab25f","name":"theta_plus_one of basis_map_A007318 of Triangle read by rows: coefficients of 1; 1(X+2); 1(X+2)(X+3); 1(X+2)(X+3)(X+4); ....","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"],["3","4"],["12","34","18"],["60","266","294","96"],["360","2196","3759","2536","600"],["2520","19692","46077","47824","23220","4320"],["20160","192816","575382","812644","604220","229608","35280"],["181440","2057904","7492230","13451204","13564420","7866984","2460528","322560"],["1814400","23844960","102704004","224418800","286970245","225165576","107208612","28533888","3265920"],["19958400","298582560","1487418444","3837993520","5961907695","5920464276","3806226452","1539114048","356931360","36288000"],["239500800","4022075520","22775633568","67871392752","124315303240","149742752886","121654008966","66319072544","23329712664","4797950400","439084800"],["3113510400","58035000960","368480595744","1246738242864","2634169833400","3736796154174","3677900656962","2530545780320","1198593743976","373436190720","69041047680","5748019200"],["43589145600","893441283840","6290410859136","23841332392992","57137874114140","93408195559476","108165350869107","90070736399344","53789544452844","22541127416400","6306712194816","1059599854080","80951270400"],["653837184000","14622115334400","113118429847680","475040988598752","1274291428589460","2361039218509692","3148147457174713","3081706659895872","2225457265082328","1174843158290240","441734045134800","112214693150208","17284866842880","1220496076800"],["10461394944000","253568960870400","2138892472586880","9863024414533632","29295681393657960","60713873210765352","91688545638715018","103275548681329032","87611116127060088","55888444950682640","26452656833945760","9023810374427328","2100046633176960","298745016998400","19615115520000"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","7","64","716","9451","143653","2470110","47397270","1003926405","23264884755","585506488140","15902598776400","463628865812175","14441374659711825","478634031219213450"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["3","4"]],"recurrence":{"denominator":{"coeffs":[["0"],["1"]]},"terms":[{"coeff":{"coeffs":[["-1","0","0"],["-3","-2","0"],["-3","-4","-1"],["-1","-2","-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["1","0"],["4","2"],["2","2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":186111,"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-9b3006716b16b890"]},{"id":186112,"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-9b3006716b16b890"]},{"id":186113,"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-9b3006716b16b890"]},{"id":186114,"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-9b3006716b16b890"]}],"region_exact":{},"incoming_edges":[{"id":"e-9b3006716b16b890","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-b6060bc044d672a9","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":"93612bfb8a3cf35f24ae6793ba9fbc1dbbd4cc2ac4997f9ce0cec740d26fba62","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-b8883330fe0b24f8","role":"sequence","name":"basis_map_A007318 of Triangle read by rows: coefficients of 1; 1(X+2); 1(X+2)(X+3); 1(X+2)(X+3)(X+4); ....","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:        3,         4;\n  2:       12,        34,         18;\n  3:       60,       266,        294,         96;\n  4:      360,      2196,       3759,       2536,        600;\n  5:     2520,     19692,      46077,      47824,      23220,       4320;\n  6:    20160,    192816,     575382,     812644,     604220,     229608,      35280;\n  7:   181440,   2057904,    7492230,   13451204,   13564420,    7866984,    2460528,     322560;\n  8:  1814400,  23844960,  102704004,  224418800,  286970245,  225165576,  107208612,   28533888,   3265920;\n  9: 19958400, 298582560, 1487418444, 3837993520, 5961907695, 5920464276, 3806226452, 1539114048, 356931360, 36288000;\n","recurrence_sentence":"The row polynomials satisfy n*P(n,x) = -(n^3+3*n^2+3*n+1+(2*n^3+4*n^2+2*n)*x+(n^3+n^2)*x^2)*P(n-2,x) + (2*n^2+4*n+1+(2*n^2+2*n)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 4*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3 + 4*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^3 + 3*n^2 + 3*n + 1 + (2*n^3 + 4*n^2 + 2*n)*x + (n^3 + n^2)*x^2)*P[n - 2, x] + (2*n^2 + 4*n + 1 + (2*n^2 + 2*n)*x)*P[n - 1, x])/(n)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^3 + 3*n^2 + 3*n + 1 + (2*n^3 + 4*n^2 + 2*n)*x + (n^3 + n^2)*x^2)*v[i - 2] + (2*n^2 + 4*n + 1 + (2*n^2 + 2*n)*x)*v[i - 1])/(n)); 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-2feb6dc01f424b0a","steps":["basis A007318","\u03b8+1"],"edges":["e-5281dccfa5f22cf2","e-9b3006716b16b890"],"text":"A145324 \u2192 basis A007318 \u2192 \u03b8+1"},"headline":"A145324 \u2192 basis A007318 \u2192 \u03b8+1","chain":null}