{"id":"n-9f8bb2102c91e8bb","revision":0,"first_index":0,"fingerprint":"f6-2c953ff2a550d931","name":"theta_plus_one_squared of basis_map_A013611 of Triangle T(n, k) read by rows: T(n, k) = Sum_{j=0..n} binomial(j, n-k) * |Stirling1(n, n-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":false,"rows":[["1"],["1","16"],["3","80","288"],["15","544","3600","6144"],["105","4768","44784","145408","153600"],["945","51312","617616","2900992","5926400","4423680"],["10395","655152","9564624","58262528","173977600","253476864","144506880"],["135135","9681216","165614688","1233518592","4810694400","10309976064","11538874368","5284823040"],["2027025","162515520","3181267872","27924414464","133836345600","376326438912","622139850752","561550196736","214035333120"],["34459425","3054655440","67245310944","678316189696","3845966188800","13335625617408","29016372969472","38799623061504","29224231501824","9512681472000"],["654729075","63551961360","1552579897536","17669896382464","115468119443200","474904539205632","1277361814478848","2253150283104256","2519491316023296","1623912873984000","460413783244800"],["13749310575","1449823505760","38895822022896","492514073794560","3639635884601600","17289055371239424","55266103285370880","120725357365035008","178381624026857472","170963427026534400","96125810859048960","24108939922636800"],["316234143225","35985808262880","1051184845148688","14646933843228672","120650180880428800","649219905882169344","2400671993151172608","6241529531490893824","11436842812543598592","14502921726905548800","12140449491459244032","6045593260020203520","1358136948975206400"],["7905853580625","965421908362800","30489960695474160","463349545098906624","4206317896773049600","25264783191380742144","105967142056217300992","319087960597621374976","696691732604834414592","1096790112619882086400","1216035072343378231296","902443019864528388096","402902545731724247040","81906105230504755200"],["213458046676875","27837302727855600","944872099231515120","15545882693575461888","154113668409026668800","1021346621363058757632","4786850754327554805760","16366114469390528282624","41426035987686944145408","77779675047984627712000","107151444984394422419456","105408166688218316537856","70188929429186392621056","28378673965196441026560","5265392479103877120000"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","17","371","10303","348665","13920945","640454043","33353317503","1939158390001","124483151426513","8743989412454467","667032827209958335","54911750272039249385","4851789604970400914545","457943262900814771606635"],"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"],["1","16"]],"recurrence":{"denominator":{"coeffs":[["-1"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","0","0"],["-9","-48","144"],["12","80","-288"],["-4","-32","144"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0","0","0"],["0","0","48","64"],["0","0","-80","-128"],["0","0","32","64"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["1","0"],["-6","-16"],["4","16"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","0"],["0","0","16"],["0","0","-16"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":190915,"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-d7189a30134fce41"]},{"id":190916,"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-d7189a30134fce41"]},{"id":190917,"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-d7189a30134fce41"]},{"id":190918,"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-d7189a30134fce41"]}],"region_exact":{},"incoming_edges":[{"id":"e-d7189a30134fce41","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-9f8bb2102c91e8bb","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":"273cc04895e2bd4e912dee0bf945759cec3dd49e42a24e546b596553a828ae0f","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-054881b8a1092a7d","role":"sequence","name":"basis_map_A013611 of Triangle T(n, k) read by rows: T(n, k) = Sum_{j=0..n} binomial(j, n-k) * |Stirling1(n, n-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":[],"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:        1,         16;\n  2:        3,         80,         288;\n  3:       15,        544,        3600,         6144;\n  4:      105,       4768,       44784,       145408,        153600;\n  5:      945,      51312,      617616,      2900992,       5926400,        4423680;\n  6:    10395,     655152,     9564624,     58262528,     173977600,      253476864,      144506880;\n  7:   135135,    9681216,   165614688,   1233518592,    4810694400,    10309976064,    11538874368,     5284823040;\n  8:  2027025,  162515520,  3181267872,  27924414464,  133836345600,   376326438912,   622139850752,   561550196736,   214035333120;\n  9: 34459425, 3054655440, 67245310944, 678316189696, 3845966188800, 13335625617408, 29016372969472, 38799623061504, 29224231501824, 9512681472000;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = -(4*n^3-12*n^2+9*n+(32*n^3-80*n^2+48*n)*x-(144*n^3-288*n^2+144*n)*x^2)*P(n-2,x) + ((32*n^3-80*n^2+48*n)*x^2+(64*n^3-128*n^2+64*n)*x^3)*d/dx P(n-2,x) + (4*n^2-6*n+1+(16*n^2-16*n)*x)*P(n-1,x) - (16*n^2-16*n)*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 16*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 16*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(4*n^3 - 12*n^2 + 9*n + (32*n^3 - 80*n^2 + 48*n)*x - (144*n^3 - 288*n^2 + 144*n)*x^2)*P[n - 2, x] + ((32*n^3 - 80*n^2 + 48*n)*x^2 + (64*n^3 - 128*n^2 + 64*n)*x^3)*D[P[n - 2, x], x] + (4*n^2 - 6*n + 1 + (16*n^2 - 16*n)*x)*P[n - 1, x] - (16*n^2 - 16*n)*x^2*D[P[n - 1, x], x])/(n - 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 16*x; for(n = 2, N, my(i = n + 1); v[i] = (-(4*n^3 - 12*n^2 + 9*n + (32*n^3 - 80*n^2 + 48*n)*x - (144*n^3 - 288*n^2 + 144*n)*x^2)*v[i - 2] + ((32*n^3 - 80*n^2 + 48*n)*x^2 + (64*n^3 - 128*n^2 + 64*n)*x^3)*deriv(v[i - 2], x) + (4*n^2 - 6*n + 1 + (16*n^2 - 16*n)*x)*v[i - 1] - (16*n^2 - 16*n)*x^2*deriv(v[i - 1], x))/(n - 1)); 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-849de7332df28657","steps":["basis A013611","(\u03b8+1)\u00b2"],"edges":["e-3d3e7193ca830199","e-d7189a30134fce41"],"text":"A088996 \u2192 basis A013611 \u2192 (\u03b8+1)\u00b2"},"headline":"A088996 \u2192 basis A013611 \u2192 (\u03b8+1)\u00b2","chain":null}