{"id":"n-8aa04e8348532867","revision":0,"first_index":1,"fingerprint":"f6-915ca4b22aed877c","name":"theta_plus_one_squared of basis_map_A008277 of basis-map coefficients of A scaled Hermite triangle.","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","12","9"],["1","28","54","16"],["1","60","225","160","25"],["1","124","810","1040","375","36"],["1","252","2709","5600","3500","756","49"],["1","508","8694","27216","26250","9576","1372","64"],["1","1020","27225","124320","173775","95256","22638","2304","81"],["1","2044","83970","545680","1063125","821772","288120","48000","3645","100"],["1","4092","256509","2332000","6168250","6461532","3135363","760320","93555","5500","121"],["1","8188","778734","9784016","34485000","47651472","30742404","10177728","1804275","170500","7986","144"],["1","16380","2354625","40520480","187712525","335567232","280055776","121575168","29119662","3932500","294151","11232","169"],["1","32764","7100730","166267920","1001875875","2283709428","2417134720","1338388480","415945530","75275200","7993986","484848","15379","196"],["1","65532","21375909","677695200","5269173000","15144957828","20028325317","13864181760","5437407690","1266265000","179016838","15331680","768950","20580","225"],["1","131068","64275174","2748782416","27404763750","98457356088","160812247596","137072899392","66483524250","19375499000","3501365868","397024992","27989780","1179920","27000","256"],["1","262140","193120425","11109396640","141318791275","630206996328","1259697134514","1306623681792","771834606543","275833415000","61959378338","8931214656","829820082","48980400","1759500","34816","289"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["22","99","471","2386","12867","73681","446620","2856457","19217243","135610448","1001159901","7714225057","61904585510","516347066551","4468588592739"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one_squared","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1","12","9"],["1","28","54","16"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","-4","9"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","-2","4","4"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0","-1","4"]]},"deriv_order":2,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["1","4"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","2","-4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":178738,"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-366bc73988f49f51"]},{"id":178739,"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-366bc73988f49f51"]},{"id":178740,"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-366bc73988f49f51"]},{"id":178741,"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-366bc73988f49f51"]}],"region_exact":{},"incoming_edges":[{"id":"e-366bc73988f49f51","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"euler_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"2\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-8aa04e8348532867","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":"b92b7c27ae96f7209fbe2693b2b9cf4a2f8c5d79475fae0351b7202dc44c7097","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-cae59b180fd3bbb3","role":"sequence","name":"basis_map_A008277 of basis-map coefficients of A scaled Hermite triangle.","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":[{"id":"l-18fc79dd0cff7737","kind":"fingerprint_related","node_a_id":"n-0abf36e1d724d74f","node_b_id":"n-8aa04e8348532867","fingerprint":"f6-915ca4b22aed877c","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-915ca4b22aed877c"},"other_id":"n-0abf36e1d724d74f","other":{"id":"n-0abf36e1d724d74f","name":"theta_plus_one_squared of Triangle of Stirling numbers of the second kind, S2(n,k), n >= 1, 1 <= k <= n.","first_index":1,"fingerprint":"f6-915ca4b22aed877c","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","4"],["1","12","9"]],"card_rows":[["1"],["1","4"],["1","12","9"],["1","28","54","16"]],"row_sums_preview":["1","5","22","99","471","2386","12867","73681"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3cb22164e2f83fe1","steps":["(\u03b8+1)\u00b2"],"edges":["e-72cd2be7747bf330"],"text":"A008277 \u2192 (\u03b8+1)\u00b2"},"headline":"A008277 \u2192 (\u03b8+1)\u00b2","chain":null}},{"id":"l-2c1a24babba979e9","kind":"fingerprint_related","node_a_id":"n-8aa04e8348532867","node_b_id":"n-483bdd27270bc439","fingerprint":"f6-915ca4b22aed877c","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-915ca4b22aed877c"},"other_id":"n-483bdd27270bc439","other":{"id":"n-483bdd27270bc439","name":"theta_plus_one_squared of basis_map_A008277 of Infinite lower triangular matrix with 2^k in the right diagonal and the rest zeros. Triangle, T(n,k), n zeros followed by the term 2^k. Triangle by columns, (2^k, 0, 0, 0, ...).","first_index":0,"fingerprint":"f6-915ca4b22aed877c","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","8"],["4","48","36"]],"card_rows":[["1"],["2","8"],["4","48","36"],["8","224","432","128"]],"row_sums_preview":["1","10","88","792","7536","76352","823488","9431168"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ae7910bbe46f8a41","steps":["basis A008277","(\u03b8+1)\u00b2"],"edges":["e-dbb6853ea73a13a7","e-9bc8f60d61d4cce4"],"text":"A130123 \u2192 basis A008277 \u2192 (\u03b8+1)\u00b2"},"headline":"A130123 \u2192 basis A008277 \u2192 (\u03b8+1)\u00b2","chain":null}},{"id":"l-2d42bf72da1aa6ab","kind":"fingerprint_related","node_a_id":"n-8aa04e8348532867","node_b_id":"n-1e68a1963f2d6543","fingerprint":"f6-915ca4b22aed877c","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-915ca4b22aed877c"},"other_id":"n-1e68a1963f2d6543","other":{"id":"n-1e68a1963f2d6543","name":"theta_plus_one_squared of basis_map_A008277 of basis-map coefficients of Triangle of coefficients in expansion of x^n in terms of Laguerre polynomials L_n(x).","first_index":1,"fingerprint":"f6-915ca4b22aed877c","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":0,"preview_rows":[["1"],["-1","-4"],["2","24","18"]],"card_rows":[["1"],["-1","-4"],["2","24","18"],["-6","-168","-324","-96"]],"row_sums_preview":["1","-5","44","-594","11304","-286320","9264240","-371352240"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-cf1765de6ac2bd4b","steps":["basis A008277","(\u03b8+1)\u00b2"],"edges":["e-2b559797932c7395","e-f5050840619a384e"],"text":"basis-map coefficients of A021012 \u2192 basis A008277 \u2192 (\u03b8+1)\u00b2"},"headline":"basis-map coefficients of A021012 \u2192 basis A008277 \u2192 (\u03b8+1)\u00b2","chain":null}},{"id":"l-8cdd0b32cb47127a","kind":"fingerprint_related","node_a_id":"n-9603c40887c5e6f8","node_b_id":"n-8aa04e8348532867","fingerprint":"f6-915ca4b22aed877c","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-915ca4b22aed877c"},"other_id":"n-9603c40887c5e6f8","other":{"id":"n-9603c40887c5e6f8","name":"theta_plus_one_squared of basis_map_A008277 of a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.","first_index":0,"fingerprint":"f6-915ca4b22aed877c","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","4"],["1","12","9"]],"card_rows":[["1"],["1","4"],["1","12","9"],["1","28","54","16"]],"row_sums_preview":["1","5","22","99","471","2386","12867","73681"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-e65f982fa1b436c5","steps":["D+1","(\u03b8+1)\u00b2"],"edges":["e-7b7980dbcf1b9283","e-f7e1aad7ae359315"],"text":"A048993 \u2192 D+1 \u2192 (\u03b8+1)\u00b2"},"headline":"A048993 \u2192 D+1 \u2192 (\u03b8+1)\u00b2","chain":null}},{"id":"l-b9f5d5fcddb6191e","kind":"fingerprint_related","node_a_id":"n-8aa04e8348532867","node_b_id":"n-435207fe015f8c82","fingerprint":"f6-915ca4b22aed877c","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-915ca4b22aed877c"},"other_id":"n-435207fe015f8c82","other":{"id":"n-435207fe015f8c82","name":"reversal of theta_plus_one_squared of Triangle of Stirling numbers of the second kind, S2(n,k), n >= 1, 1 <= k <= n.","first_index":1,"fingerprint":"f6-915ca4b22aed877c","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["4","1"],["9","12","1"]],"card_rows":[["1"],["4","1"],["9","12","1"],["16","54","28","1"]],"row_sums_preview":["1","5","22","99","471","2386","12867","73681"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3cb22164e2f83fe1","steps":["(\u03b8+1)\u00b2","reverse"],"edges":["e-72cd2be7747bf330","e-dbd7f57a44e504f6"],"text":"A008277 \u2192 (\u03b8+1)\u00b2 \u2192 reverse"},"headline":"A008277 \u2192 (\u03b8+1)\u00b2 \u2192 reverse","chain":null}},{"id":"l-d1d6c3a5196fd506","kind":"fingerprint_related","node_a_id":"n-dce4c0707068cc4f","node_b_id":"n-8aa04e8348532867","fingerprint":"f6-915ca4b22aed877c","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-915ca4b22aed877c"},"other_id":"n-dce4c0707068cc4f","other":{"id":"n-dce4c0707068cc4f","name":"theta_plus_one of derivative of Triangle of Stirling numbers of 2nd kind, S(n,k), n >= 0, 0 <= k <= n.","first_index":0,"fingerprint":"f6-915ca4b22aed877c","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["1"],["1","4"]],"card_rows":[["1"],["1","4"],["1","12","9"],["1","28","54","16"]],"row_sums_preview":["0","1","5","22","99","471","2386","12867"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3cb22164e2f83fe1","steps":["\u03b8+1","\u03b8+1"],"edges":["e-41f88f689ebf91d9","e-23f26c0bd195bf9a"],"text":"A008277 \u2192 \u03b8+1 \u2192 \u03b8+1"},"headline":"A008277 \u2192 \u03b8+1 \u2192 \u03b8+1","chain":null}}],"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,   12,      9;\n   2: 1,   28,     54,      16;\n   3: 1,   60,    225,     160,       25;\n   4: 1,  124,    810,    1040,      375,       36;\n   5: 1,  252,   2709,    5600,     3500,      756,       49;\n   6: 1,  508,   8694,   27216,    26250,     9576,     1372,       64;\n   7: 1, 1020,  27225,  124320,   173775,    95256,    22638,     2304,      81;\n   8: 1, 2044,  83970,  545680,  1063125,   821772,   288120,    48000,    3645,    100;\n   9: 1, 4092, 256509, 2332000,  6168250,  6461532,  3135363,   760320,   93555,   5500,  121;\n  10: 1, 8188, 778734, 9784016, 34485000, 47651472, 30742404, 10177728, 1804275, 170500, 7986, 144;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(4*x-9*x^2)*P(n-2,x) - (2*x-4*x^2-4*x^3)*d/dx P(n-2,x) - (x^2-4*x^3)*d^2/dx^2 P(n-2,x) + (1+4*x)*P(n-1,x) + (2*x-4*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1 + 12*x + 9*x^2, P(2,x) = 1 + 28*x + 54*x^2 + 16*x^3.","mathematica":"P[1, x_] := 1 + 12*x + 9*x^2;\nP[2, x_] := 1 + 28*x + 54*x^2 + 16*x^3;\nP[n_, x_] := P[n, x] = Expand[-(4*x - 9*x^2)*P[n - 2, x] - (2*x - 4*x^2 - 4*x^3)*D[P[n - 2, x], x] - (x^2 - 4*x^3)*D[P[n - 2, x], {x, 2}] + (1 + 4*x)*P[n - 1, x] + (2*x - 4*x^2)*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 + 12*x + 9*x^2; v[2] = 1 + 28*x + 54*x^2 + 16*x^3; for(n = 3, N, my(i = n); v[i] = -(4*x - 9*x^2)*v[i - 2] - (2*x - 4*x^2 - 4*x^3)*deriv(v[i - 2], x) - (x^2 - 4*x^3)*deriv(deriv(v[i - 2], x), x) + (1 + 4*x)*v[i - 1] + (2*x - 4*x^2)*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-5fe240122e98ffb3","steps":["basis A008277","(\u03b8+1)\u00b2"],"edges":["e-66762082773dd3e0","e-366bc73988f49f51"],"text":"basis-map coefficients of A112227 \u2192 basis A008277 \u2192 (\u03b8+1)\u00b2"},"headline":"basis-map coefficients of A112227 \u2192 basis A008277 \u2192 (\u03b8+1)\u00b2","chain":null}
