{"id":"n-483bdd27270bc439","revision":0,"first_index":0,"fingerprint":"f6-915ca4b22aed877c","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, ...).","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","8"],["4","48","36"],["8","224","432","128"],["16","960","3600","2560","400"],["32","3968","25920","33280","12000","1152"],["64","16128","173376","358400","224000","48384","3136"],["128","65024","1112832","3483648","3360000","1225728","175616","8192"],["256","261120","6969600","31825920","44486400","24385536","5795328","589824","20736"],["512","1046528","42992640","279388160","544320000","420747264","147517440","24576000","1866240","51200"],["1024","4190208","262665216","2387968000","6316288000","6616608768","3210611712","778567680","95800320","5632000","123904"],["2048","16769024","1594847232","20037664768","70625280000","97590214656","62960443392","20843986944","3695155200","349184000","16355328","294912"],["4096","67092480","9644544000","165971886080","768870502400","1374483382272","1147108458496","497971888128","119274135552","16107520000","1204842496","46006272","692224"],["8192","268402688","58169180160","1362066800640","8207367168000","18708147634176","19801167626240","10964078428160","3407425781760","616654438400","65486733312","3971874816","125984768","1605632"],["16384","1073676288","350222893056","11103358156800","86330130432000","248134989053952","328144081993728","227150753955840","89086487592960","20746485760000","2933011873792","251194245120","12598476800","337182720","3686400"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","10","88","792","7536","76352","823488","9431168","114334720","1462505984","19678456832","277730197504","4100750954496","63194931666944","1014244728995840"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one_squared","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["2","8"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","-16","36"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","-8","16","16"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0","-4","16"]]},"deriv_order":2,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["2","8"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","4","-8"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":181773,"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-9bc8f60d61d4cce4"]},{"id":181774,"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-9bc8f60d61d4cce4"]},{"id":181775,"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-9bc8f60d61d4cce4"]},{"id":181776,"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-9bc8f60d61d4cce4"]}],"region_exact":{},"incoming_edges":[{"id":"e-9bc8f60d61d4cce4","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-483bdd27270bc439","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":"42700ff6c0a88dc231542c09fbccd25e927f8d4a23c835437b4d08f77942ae72","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-95ac664928981c03","role":"sequence","name":"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, ...).","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-01b46d1bc178a6d9","kind":"fingerprint_related","node_a_id":"n-dce4c0707068cc4f","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-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}},{"id":"l-02ddacbf7f11f9b9","kind":"fingerprint_related","node_a_id":"n-435207fe015f8c82","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-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-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-8aa04e8348532867","other":{"id":"n-8aa04e8348532867","name":"theta_plus_one_squared of basis_map_A008277 of basis-map coefficients of A scaled Hermite triangle.","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","12","9"],["1","28","54","16"],["1","60","225","160","25"]],"card_rows":[["1","12","9"],["1","28","54","16"],["1","60","225","160","25"],["1","124","810","1040","375","36"]],"row_sums_preview":["22","99","471","2386","12867","73681","446620","2856457"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"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}},{"id":"l-59b670a392ef942d","kind":"fingerprint_related","node_a_id":"n-9603c40887c5e6f8","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-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-cb1dbfc0ab0ab3f2","kind":"fingerprint_related","node_a_id":"n-483bdd27270bc439","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-fb6d9e0b6ac36748","kind":"fingerprint_related","node_a_id":"n-0abf36e1d724d74f","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-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}}],"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,       8;\n  2:   4,      48,       36;\n  3:   8,     224,      432,       128;\n  4:  16,     960,     3600,      2560,       400;\n  5:  32,    3968,    25920,     33280,     12000,      1152;\n  6:  64,   16128,   173376,    358400,    224000,     48384,      3136;\n  7: 128,   65024,  1112832,   3483648,   3360000,   1225728,    175616,     8192;\n  8: 256,  261120,  6969600,  31825920,  44486400,  24385536,   5795328,   589824,   20736;\n  9: 512, 1046528, 42992640, 279388160, 544320000, 420747264, 147517440, 24576000, 1866240, 51200;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(16*x-36*x^2)*P(n-2,x) - (8*x-16*x^2-16*x^3)*d/dx P(n-2,x) - (4*x^2-16*x^3)*d^2/dx^2 P(n-2,x) + (2+8*x)*P(n-1,x) + (4*x-8*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 8*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + 8*x;\nP[n_, x_] := P[n, x] = Expand[-(16*x - 36*x^2)*P[n - 2, x] - (8*x - 16*x^2 - 16*x^3)*D[P[n - 2, x], x] - (4*x^2 - 16*x^3)*D[P[n - 2, x], {x, 2}] + (2 + 8*x)*P[n - 1, x] + (4*x - 8*x^2)*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = -(16*x - 36*x^2)*v[i - 2] - (8*x - 16*x^2 - 16*x^3)*deriv(v[i - 2], x) - (4*x^2 - 16*x^3)*deriv(deriv(v[i - 2], x), x) + (2 + 8*x)*v[i - 1] + (4*x - 8*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 - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"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}