{"id":"n-d665c3dd5d9a2622","revision":0,"first_index":0,"fingerprint":"f6-950975dd37953e66","name":"theta_plus_one_squared of basis_map_A007318 of Triangle of coefficients in expansion of (1+6x)^n.","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"],["7","24"],["49","336","324"],["343","3528","6804","3456"],["2401","32928","95256","96768","32400"],["16807","288120","1111320","1693440","1134000","279936"],["117649","2420208","11668860","23708160","23814000","11757312","2286144"],["823543","19765032","114354828","290424960","388962000","288054144","112021056","17915904"],["5764801","158120256","1067311728","3252759552","5445468000","5377010688","3136589568","1003290624","136048896"],["40353607","1245197016","9605805552","34153975296","68612896800","84687918336","65868380928","31603654656","8571080448","1007769600"],["282475249","9684865680","84050798580","341539752960","800483796000","1185630856704","1152696666240","737418608640","299987815680","70543872000","7316407296"],["1977326743","74573465736","719101276740","3287320122240","8805321756000","15215595994368","17751528660096","14195308216320","7699687269120","2715939072000","563363361792","52242776064"],["13841287201","569470101984","6040450724616","30681654474240","92455878438000","182587151932416","248521401241344","238481178034176","161693432651520","76046294016000","23661261195264","4388393189376","367876214784"],["96889010407","4318481606712","49971001449096","279203055715584","934831659762000","2076928853231232","3230778216137472","3616964533518336","2942820474257664","1730053188864000","717724922923008","199671890116608","33476735545344","2559896027136"],["678223072849","32554707496752","408096511834284","2487445405466112","9161350265667600","22615447512962304","39577033147684032","50637503469256704","48066067746208512","33909042501734400","17584260611613696","6522615077142528","1640360041721856","250869810659328","17631936921600"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","31","709","14131","259753","4523623","75772333","1232321467","19582364113","305397032239","4689635915029","71081959297219","1065508283500921","15819399798164599","232910956969442557"],"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"]],"recurrence":{"denominator":{"coeffs":[["49","-42","0"],["0","126","0"],["0","0","36"]]},"terms":[{"coeff":{"coeffs":[["343","882","756","216"],["0","882","1260","432"],["0","0","252","216"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":187442,"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-f92d9ffdafb6b8cc"]},{"id":187443,"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-f92d9ffdafb6b8cc"]},{"id":187444,"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-f92d9ffdafb6b8cc"]},{"id":187445,"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-f92d9ffdafb6b8cc"]}],"region_exact":{},"incoming_edges":[{"id":"e-f92d9ffdafb6b8cc","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-d665c3dd5d9a2622","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":"395e795ab7fac81e7d9064527c25ef39d7e91a6ecbe97be80c99ddedc078fbfa","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-bc97c3236a99a08c","role":"sequence","name":"basis_map_A007318 of Triangle of coefficients in expansion of (1+6x)^n.","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-1a34765d941a08bd","kind":"fingerprint_related","node_a_id":"n-d665c3dd5d9a2622","node_b_id":"n-69caac66622b328f","fingerprint":"f6-950975dd37953e66","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-950975dd37953e66"},"other_id":"n-69caac66622b328f","other":{"id":"n-69caac66622b328f","name":"theta_plus_one_squared of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*9^j.","first_index":0,"fingerprint":"f6-950975dd37953e66","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["21","72"],["441","3024","2916"]],"card_rows":[["1"],["21","72"],["441","3024","2916"],["9261","95256","183708","93312"]],"row_sums_preview":["1","93","6381","381537","21039993","1099240389","55238030757","2695087048329"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-f485cd65a8f8089f","steps":["basis A013609","(\u03b8+1)\u00b2"],"edges":["e-f8a7b8e38f7708ee","e-02cb642d98626197"],"text":"A038335 \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2"},"headline":"A038335 \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2","chain":null}},{"id":"l-a41c908b404339ee","kind":"fingerprint_related","node_a_id":"n-d665c3dd5d9a2622","node_b_id":"n-f7ef7289ee552610","fingerprint":"f6-950975dd37953e66","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-950975dd37953e66"},"other_id":"n-f7ef7289ee552610","other":{"id":"n-f7ef7289ee552610","name":"theta_plus_one_squared of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*6^j.","first_index":0,"fingerprint":"f6-950975dd37953e66","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["196","1344","1296"],["2744","28224","54432","27648"],["38416","526848","1524096","1548288","518400"]],"card_rows":[["196","1344","1296"],["2744","28224","54432","27648"],["38416","526848","1524096","1548288","518400"],["537824","9219840","35562240","54190080","36288000","8957952"]],"row_sums_preview":["2836","113048","4156048","144755936","4849429312","157737147776","5013085212928","156363280506368"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c11cb6bca0e3e7dc","steps":["basis A013609","(\u03b8+1)\u00b2"],"edges":["e-65e466871b01ed5e","e-ac130b6999234204"],"text":"A038284 \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2"},"headline":"A038284 \u2192 basis A013609 \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:        7,         24;\n  2:       49,        336,        324;\n  3:      343,       3528,       6804,        3456;\n  4:     2401,      32928,      95256,       96768,       32400;\n  5:    16807,     288120,    1111320,     1693440,     1134000,      279936;\n  6:   117649,    2420208,   11668860,    23708160,    23814000,    11757312,     2286144;\n  7:   823543,   19765032,  114354828,   290424960,   388962000,   288054144,   112021056,    17915904;\n  8:  5764801,  158120256, 1067311728,  3252759552,  5445468000,  5377010688,  3136589568,  1003290624,  136048896;\n  9: 40353607, 1245197016, 9605805552, 34153975296, 68612896800, 84687918336, 65868380928, 31603654656, 8571080448, 1007769600;\n","recurrence_sentence":"The row polynomials satisfy (49+(126*n-42)*x+36*n^2*x^2)*P(n,x) = (343+(882*n+882)*x+(252*n^2+1260*n+756)*x^2+(216*n^2+432*n+216)*x^3)*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[((343 + (882*n + 882)*x + (252*n^2 + 1260*n + 756)*x^2 + (216*n^2 + 432*n + 216)*x^3)*P[n - 1, x])/(49 + (126*n - 42)*x + 36*n^2*x^2)]];\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] = ((343 + (882*n + 882)*x + (252*n^2 + 1260*n + 756)*x^2 + (216*n^2 + 432*n + 216)*x^3)*v[i - 1])/(49 + (126*n - 42)*x + 36*n^2*x^2)); 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-efce1b6c07b70e1a","steps":["x \u21a6 x+1","(\u03b8+1)\u00b2"],"edges":["e-3efa57e183892da5","e-f92d9ffdafb6b8cc"],"text":"A013613 \u2192 x \u21a6 x+1 \u2192 (\u03b8+1)\u00b2"},"headline":"A013613 \u2192 x \u21a6 x+1 \u2192 (\u03b8+1)\u00b2","chain":null}
