{"id":"n-0afee164e61c22a7","revision":1,"first_index":0,"fingerprint":"f6-14ef8ed0e797c5a3","name":"theta_plus_one of Triangle of coefficients in expansion of (1+4x)^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":true,"rows":[["1"],["1","8"],["1","16","48"],["1","24","144","256"],["1","32","288","1024","1280"],["1","40","480","2560","6400","6144"],["1","48","720","5120","19200","36864","28672"],["1","56","1008","8960","44800","129024","200704","131072"],["1","64","1344","14336","89600","344064","802816","1048576","589824"],["1","72","1728","21504","161280","774144","2408448","4718592","5308416","2621440"],["1","80","2160","30720","268800","1548288","6021120","15728640","26542080","26214400","11534336"],["1","88","2640","42240","422400","2838528","13246464","43253760","97320960","144179200","126877696","50331648"],["1","96","3168","56320","633600","4866048","26492928","103809024","291962880","576716800","761266176","603979776","218103808"],["1","104","3744","73216","915200","7907328","49201152","224919552","759103488","1874329600","3298820096","3925868544","2835349504","939524096"],["1","112","4368","93184","1281280","12300288","86102016","449839104","1771241472","5248122880","11545870336","18320719872","19847446528","13153337344","4026531840"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","9","65","425","2625","15625","90625","515625","2890625","16015625","87890625","478515625","2587890625","13916015625","74462890625"],"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":[["1","0"],["0","4"]]},"terms":[{"coeff":{"coeffs":[["1","8","16"],["0","4","16"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":165762,"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-c46c299bd4ab7a25"]},{"id":165763,"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-c46c299bd4ab7a25"]},{"id":165764,"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-c46c299bd4ab7a25"]},{"id":165765,"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-c46c299bd4ab7a25"]}],"region_exact":{},"incoming_edges":[{"id":"e-c46c299bd4ab7a25","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-0afee164e61c22a7","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":"d3b3bd3f3898f79037cc47efb76de7d45d3e2b8d4ae2349ba468e90f4af8bdcc","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-2e0409c0014ec240","role":"sequence","name":"Triangle of coefficients in expansion of (1+4x)^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-eb7561f7babe3333","kind":"oeis_match","node_a_id":"n-0afee164e61c22a7","node_b_id":null,"fingerprint":null,"oeis_id":"A081040","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A081040","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A081040","relation":"match_row_sums"}],"bfiles":[{"node_id":"n-0afee164e61c22a7","oeis_id":"A081040","target":"row_sums","oeis_offset":0,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":24,"oeis_terms_available":24,"term_count":200,"raw_size":15254}],"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,  8;\n  2: 1, 16,   48;\n  3: 1, 24,  144,   256;\n  4: 1, 32,  288,  1024,   1280;\n  5: 1, 40,  480,  2560,   6400,   6144;\n  6: 1, 48,  720,  5120,  19200,  36864,   28672;\n  7: 1, 56, 1008,  8960,  44800, 129024,  200704,  131072;\n  8: 1, 64, 1344, 14336,  89600, 344064,  802816, 1048576,  589824;\n  9: 1, 72, 1728, 21504, 161280, 774144, 2408448, 4718592, 5308416, 2621440;\n","recurrence_sentence":"The row polynomials satisfy (1+4*n*x)*P(n,x) = (1+(4*n+8)*x+(16*n+16)*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[((1 + (4*n + 8)*x + (16*n + 16)*x^2)*P[n - 1, x])/(1 + 4*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] = ((1 + (4*n + 8)*x + (16*n + 16)*x^2)*v[i - 1])/(1 + 4*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-2e0409c0014ec240","steps":["\u03b8+1"],"edges":["e-c46c299bd4ab7a25"],"text":"A013611 \u2192 \u03b8+1"},"headline":"A013611 \u2192 \u03b8+1","chain":null}
