{"id":"n-14677aff3512ef46","revision":0,"first_index":0,"fingerprint":"f6-604d756e1453b62f","name":"theta_plus_one of basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*6^(i-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":true,"rows":[["1"],["7","8"],["49","112","48"],["343","1176","1008","256"],["2401","10976","14112","7168","1280"],["16807","96040","164640","125440","44800","6144"],["117649","806736","1728720","1756160","940800","258048","28672"],["823543","6588344","16941456","21512960","15366400","6322176","1404928","131072"],["5764801","52706752","158120256","240945152","215129600","118013952","39337984","7340032","589824"],["40353607","415065672","1423082304","2529924096","2710632960","1858719744","826097664","231211008","37158912","2621440"],["282475249","3228288560","12451970160","25299240960","31624051200","26022076416","14456709120","5394923520","1300561920","183500800","11534336"],["1977326743","24857821912","106533522480","243505194240","347864563200","333949980672","222633320448","103852277760","33381089280","7064780800","888143872","50331648"],["13841287201","189823367328","894881588832","2272715146240","3652577913600","4007399768064","3116866486272","1744718266368","701002874880","197813862400","37302042624","4227858432","218103808"],["96889010407","1439493868904","7403111325792","20681707830784","36931621126400","45584172361728","40519264321536","26461560373248","12758252322816","4500265369600","1131495292928","192367558656","19847446528","939524096"],["678223072849","10851569165584","60458742493968","184255215219712","361929887038720","496360987938816","496360987938816","370461845225472","208384787939328","88205201244160","27721634676736","6284006916096","972524879872","92073361408","4026531840"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","15","209","2783","35937","453871","5636785","69090879","837948353","10074867407","120255332241","1426558353055","16833388566049","197720987733423","2313021713643377"],"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":[["7","0"],["0","4"]]},"terms":[{"coeff":{"coeffs":[["49","56","16"],["0","28","16"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":186077,"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-e6b27e29b49c899a"]},{"id":186078,"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-e6b27e29b49c899a"]},{"id":186079,"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-e6b27e29b49c899a"]},{"id":186080,"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-e6b27e29b49c899a"]}],"region_exact":{},"incoming_edges":[{"id":"e-e6b27e29b49c899a","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-14677aff3512ef46","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":"864119d4489d5c9bdd2bc9f531cdfb255e3080b12bf4805cd449398fefbf73c7","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-ade19c5257c8a726","role":"sequence","name":"basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*6^(i-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":[{"id":"l-bb04574949aaf5d8","kind":"fingerprint_related","node_a_id":"n-14677aff3512ef46","node_b_id":"n-f37732e25746403d","fingerprint":"f6-604d756e1453b62f","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-604d756e1453b62f"},"other_id":"n-f37732e25746403d","other":{"id":"n-f37732e25746403d","name":"theta_plus_one of basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*2^j.","first_index":0,"fingerprint":"f6-604d756e1453b62f","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["196","448","192"],["2744","9408","8064","2048"],["38416","175616","225792","114688","20480"]],"card_rows":[["196","448","192"],["2744","9408","8064","2048"],["38416","175616","225792","114688","20480"],["537824","3073280","5268480","4014080","1433600","196608"]],"row_sums_preview":["836","22264","574992","14523872","360754240","8843632512","214514778368","5158332112384"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ea9b82d727693fb6","steps":["basis A013611","\u03b8+1"],"edges":["e-0eb3975e7d8d74fc","e-2c665a7ccc958e82"],"text":"A038328 \u2192 basis A013611 \u2192 \u03b8+1"},"headline":"A038328 \u2192 basis A013611 \u2192 \u03b8+1","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,         8;\n  2:       49,       112,         48;\n  3:      343,      1176,       1008,        256;\n  4:     2401,     10976,      14112,       7168,       1280;\n  5:    16807,     96040,     164640,     125440,      44800,       6144;\n  6:   117649,    806736,    1728720,    1756160,     940800,     258048,     28672;\n  7:   823543,   6588344,   16941456,   21512960,   15366400,    6322176,   1404928,    131072;\n  8:  5764801,  52706752,  158120256,  240945152,  215129600,  118013952,  39337984,   7340032,   589824;\n  9: 40353607, 415065672, 1423082304, 2529924096, 2710632960, 1858719744, 826097664, 231211008, 37158912, 2621440;\n","recurrence_sentence":"The row polynomials satisfy (7+4*n*x)*P(n,x) = (49+(28*n+56)*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[((49 + (28*n + 56)*x + (16*n + 16)*x^2)*P[n - 1, x])/(7 + 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] = ((49 + (28*n + 56)*x + (16*n + 16)*x^2)*v[i - 1])/(7 + 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-22693abd431a71f2","steps":["basis A013611","\u03b8+1"],"edges":["e-1b27bff9bafb5517","e-e6b27e29b49c899a"],"text":"A038255 \u2192 basis A013611 \u2192 \u03b8+1"},"headline":"A038255 \u2192 basis A013611 \u2192 \u03b8+1","chain":null}