{"id":"n-cea1c0a6f6924fc7","revision":0,"first_index":0,"fingerprint":"f6-710d46ba25b489cc","name":"theta_plus_one of basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*9^(i-j)*1^j.","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":true,"rows":[["1"],["10","6"],["100","120","27"],["1000","1800","810","108"],["10000","24000","16200","4320","405"],["100000","300000","270000","108000","20250","1458"],["1000000","3600000","4050000","2160000","607500","87480","5103"],["10000000","42000000","56700000","37800000","14175000","3061800","357210","17496"],["100000000","480000000","756000000","604800000","283500000","81648000","14288400","1399680","59049"],["1000000000","5400000000","9720000000","9072000000","5103000000","1837080000","428652000","62985600","5314410","196830"],["10000000000","60000000000","121500000000","129600000000","85050000000","36741600000","10716300000","2099520000","265720500","19683000","649539"],["100000000000","660000000000","1485000000000","1782000000000","1336500000000","673596000000","235758600000","57736800000","9743085000","1082565000","71449290","2125764"],["1000000000000","7200000000000","17820000000000","23760000000000","20047500000000","11547360000000","4715172000000","1385683200000","292292550000","43302600000","4286957400","255091680","6908733"],["10000000000000","78000000000000","210600000000000","308880000000000","289575000000000","187644600000000","87567480000000","30023136000000","7599606300000","1407334500000","185768154000","16580959200","898135290","22320522"],["100000000000000","840000000000000","2457000000000000","3931200000000000","4054050000000000","2918916000000000","1532430900000000","600462720000000","177324147000000","39405366000000","6501885390000","773778096000","62869470300","3124873080","71744535"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","16","247","3718","54925","799708","11510083","164111506","2321695129","32629228840","455993473039","6341490625054","87815859307813","1211500426369012","16658130862573915"],"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":[["10","0"],["0","3"]]},"terms":[{"coeff":{"coeffs":[["100","60","9"],["0","30","9"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":187193,"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-8e1e67558cbfe933"]},{"id":187194,"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-8e1e67558cbfe933"]},{"id":187195,"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-8e1e67558cbfe933"]},{"id":187196,"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-8e1e67558cbfe933"]}],"region_exact":{},"incoming_edges":[{"id":"e-8e1e67558cbfe933","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-cea1c0a6f6924fc7","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":"0c9843a54a46ea90c8cfc7393ce8d3043d159964a32339e955ad6d21afe9a667","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-6ca7f61b6ad83377","role":"sequence","name":"basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*9^(i-j)*1^j.","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-7d0af9568693b6ed","kind":"fingerprint_related","node_a_id":"n-cea1c0a6f6924fc7","node_b_id":"n-a4c468275233b706","fingerprint":"f6-710d46ba25b489cc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-710d46ba25b489cc"},"other_id":"n-a4c468275233b706","other":{"id":"n-a4c468275233b706","name":"theta_plus_one of Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*3^j.","first_index":0,"fingerprint":"f6-710d46ba25b489cc","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["100","120","27"],["1000","1800","810","108"],["10000","24000","16200","4320","405"]],"card_rows":[["100","120","27"],["1000","1800","810","108"],["10000","24000","16200","4320","405"],["100000","300000","270000","108000","20250","1458"]],"row_sums_preview":["247","3718","54925","799708","11510083","164111506","2321695129","32629228840"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-4397e742863fdbe2","steps":["\u03b8+1"],"edges":["e-1a325cc60bdaae11"],"text":"A038305 \u2192 \u03b8+1"},"headline":"A038305 \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:         10,          6;\n  2:        100,        120,         27;\n  3:       1000,       1800,        810,        108;\n  4:      10000,      24000,      16200,       4320,        405;\n  5:     100000,     300000,     270000,     108000,      20250,       1458;\n  6:    1000000,    3600000,    4050000,    2160000,     607500,      87480,      5103;\n  7:   10000000,   42000000,   56700000,   37800000,   14175000,    3061800,    357210,    17496;\n  8:  100000000,  480000000,  756000000,  604800000,  283500000,   81648000,  14288400,  1399680,   59049;\n  9: 1000000000, 5400000000, 9720000000, 9072000000, 5103000000, 1837080000, 428652000, 62985600, 5314410, 196830;\n","recurrence_sentence":"The row polynomials satisfy (10+3*n*x)*P(n,x) = (100+(30*n+60)*x+(9*n+9)*x^2)*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[((100 + (30*n + 60)*x + (9*n + 9)*x^2)*P[n - 1, x])/(10 + 3*n*x)]];\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] = ((100 + (30*n + 60)*x + (9*n + 9)*x^2)*v[i - 1])/(10 + 3*n*x)); 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-3dd184aba99388f7","steps":["basis A013610","\u03b8+1"],"edges":["e-8d10edd099560dc3","e-8e1e67558cbfe933"],"text":"A038291 \u2192 basis A013610 \u2192 \u03b8+1"},"headline":"A038291 \u2192 basis A013610 \u2192 \u03b8+1","chain":null}