{"id":"n-d4d842f5bb398cf3","revision":0,"first_index":0,"fingerprint":"f6-6ae5c48b72ea3401","name":"theta_plus_one of basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^j.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":2,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[["225","180","27"],["3375","4050","1215","108"],["50625","81000","36450","6480","405"],["759375","1518750","911250","243000","30375","1458"],["11390625","27337500","20503125","7290000","1366875","131220","5103"],["170859375","478406250","430565625","191362500","47840625","6889050","535815","17496"],["2562890625","8201250000","8611312500","4592700000","1435218750","275562000","32148900","2099520","59049"],["38443359375","138396093750","166075312500","103335750000","38750906250","9300217500","1446700500","141717600","7971615","196830"],["576650390625","2306601562500","3113912109375","2214337500000","968772656250","279006525000","54251268750","7085880000","597871125","29524500","649539"],["8649755859375","38058925781250","57088388671875","45670710937500","22835355468750","7672679437500","1790291868750","292292550000","32882911875","2435771250","107173935","2125764"],["129746337890625","622782421875000","1027590996093750","913414218750000","513795498046875","197297471250000","53708756062500","10522531800000","1479731034375","146146275000","9645654150","382637520","6908733"],["1946195068359375","10120214355468750","18216385839843750","17811577265625000","11132235791015625","4809125861718750","1496172490312500","341982283500000","57709510340625","7124630906250","626967519750","37307158200","1347202935","22320522"],["29192926025390625","163480385742187500","318786752197265625","340039202343750000","233776951611328125","112212936773437500","39274527870703125","10259468505000000","2019832861921875","299234498062500","32915794786875","2611501074000","141456308175","4687309620","71744535"],["437893890380859375","2627363342285156250","5517463018798828125","6375735043945312500","4781801282958984375","2524791077402343750","981863196767578125","288547551703125000","64923199133203125","11221293677343750","1481210765409375","146896935412500","10609223113125","527322332250","16142520375","229582512"],["6568408355712890625","42037813476562500000","94585080322265625000","117705877734375000000","95636025659179687500","55086350779687500000","23564716722421875000","7694601378750000000","1947695973996093750","384730068937500000","59248430616375000","7051052899800000","636553386787500","42185786580000","1937102445000","55099802880","731794257"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["432","8748","174960","3464208","68024448","1326476736","25713241344","495898225920","9521245937664","182093828557824","3470494144278528","65939388741292032","1249377891940270080","23613242157671104512","445278280687512256512"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["225","180","27"]],"recurrence":{"denominator":{"coeffs":[["5","2"],["0","1"]]},"terms":[{"coeff":{"coeffs":[["75","60","9"],["0","15","3"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":193071,"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-051c9abffcf73621"]},{"id":193072,"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-051c9abffcf73621"]},{"id":193073,"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-051c9abffcf73621"]},{"id":193074,"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-051c9abffcf73621"]}],"region_exact":{},"incoming_edges":[{"id":"e-051c9abffcf73621","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-d4d842f5bb398cf3","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":"e9c40d9dab4cee6a52581b59affe9c7c520764bc7727ff12758213d7e8ccddb2","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-8447c9ce68adee7a","role":"sequence","name":"basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^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-0a2f11cfcdf0b7a8","kind":"fingerprint_related","node_a_id":"n-f482e744be1cefd4","node_b_id":"n-d4d842f5bb398cf3","fingerprint":"f6-6ae5c48b72ea3401","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-6ae5c48b72ea3401"},"other_id":"n-f482e744be1cefd4","other":{"id":"n-f482e744be1cefd4","name":"theta_plus_one of Triangle whose (i,j)-th entry is 5^(i-j)*binomial(i,j).","first_index":0,"fingerprint":"f6-6ae5c48b72ea3401","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["5","2"],["25","20","3"]],"card_rows":[["1"],["5","2"],["25","20","3"],["125","150","45","4"]],"row_sums_preview":["1","7","48","324","2160","14256","93312","606528"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7b91a7a037fb3513","steps":["\u03b8+1"],"edges":["e-915fc3d2874d3b78"],"text":"A038243 \u2192 \u03b8+1"},"headline":"A038243 \u2192 \u03b8+1","chain":null}},{"id":"l-50056c2622899da5","kind":"fingerprint_related","node_a_id":"n-90989d1b314934e8","node_b_id":"n-d4d842f5bb398cf3","fingerprint":"f6-6ae5c48b72ea3401","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-6ae5c48b72ea3401"},"other_id":"n-90989d1b314934e8","other":{"id":"n-90989d1b314934e8","name":"theta_plus_one of basis_map_A013610 of Triangle of coefficients in expansion of (14 + x)^n.","first_index":0,"fingerprint":"f6-6ae5c48b72ea3401","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["15","6"],["225","180","27"]],"card_rows":[["1"],["15","6"],["225","180","27"],["3375","4050","1215","108"]],"row_sums_preview":["1","21","432","8748","174960","3464208","68024448","1326476736"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a679863398df5985","steps":["basis A013610","\u03b8+1"],"edges":["e-011a72a323f5e83e","e-0d5efd37416f29b9"],"text":"A147716 \u2192 basis A013610 \u2192 \u03b8+1"},"headline":"A147716 \u2192 basis A013610 \u2192 \u03b8+1","chain":null}},{"id":"l-91cc787014662abf","kind":"fingerprint_related","node_a_id":"n-ececab726f2e6291","node_b_id":"n-d4d842f5bb398cf3","fingerprint":"f6-6ae5c48b72ea3401","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-6ae5c48b72ea3401"},"other_id":"n-ececab726f2e6291","other":{"id":"n-ececab726f2e6291","name":"theta_plus_one of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*9^(i-j)*1^j.","first_index":0,"fingerprint":"f6-6ae5c48b72ea3401","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["10","4"],["100","80","12"]],"card_rows":[["1"],["10","4"],["100","80","12"],["1000","1200","360","32"]],"row_sums_preview":["1","14","192","2592","34560","456192","5971968","77635584"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3dd184aba99388f7","steps":["basis A013609","\u03b8+1"],"edges":["e-88ca34e9877f4f47","e-fb893e59a50c4636"],"text":"A038291 \u2192 basis A013609 \u2192 \u03b8+1"},"headline":"A038291 \u2192 basis A013609 \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:           225,            180,             27;\n  1:          3375,           4050,           1215,            108;\n  2:         50625,          81000,          36450,           6480,            405;\n  3:        759375,        1518750,         911250,         243000,          30375,          1458;\n  4:      11390625,       27337500,       20503125,        7290000,        1366875,        131220,          5103;\n  5:     170859375,      478406250,      430565625,      191362500,       47840625,       6889050,        535815,        17496;\n  6:    2562890625,     8201250000,     8611312500,     4592700000,     1435218750,     275562000,      32148900,      2099520,       59049;\n  7:   38443359375,   138396093750,   166075312500,   103335750000,    38750906250,    9300217500,    1446700500,    141717600,     7971615,     196830;\n  8:  576650390625,  2306601562500,  3113912109375,  2214337500000,   968772656250,  279006525000,   54251268750,   7085880000,   597871125,   29524500,    649539;\n  9: 8649755859375, 38058925781250, 57088388671875, 45670710937500, 22835355468750, 7672679437500, 1790291868750, 292292550000, 32882911875, 2435771250, 107173935, 2125764;\n","recurrence_sentence":"The row polynomials satisfy (5+(n+2)*x)*P(n,x) = (75+(15*n+60)*x+(3*n+9)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 225 + 180*x + 27*x^2.","mathematica":"P[0, x_] := 225 + 180*x + 27*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((75 + (15*n + 60)*x + (3*n + 9)*x^2)*P[n - 1, x])/(5 + (n + 2)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 225 + 180*x + 27*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((75 + (15*n + 60)*x + (3*n + 9)*x^2)*v[i - 1])/(5 + (n + 2)*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-615df6ea54d3bbef","steps":["x \u21a6 x+1","\u03b8+1"],"edges":["e-5083e5230277d126","e-051c9abffcf73621"],"text":"A038329 \u2192 x \u21a6 x+1 \u2192 \u03b8+1"},"headline":"A038329 \u2192 x \u21a6 x+1 \u2192 \u03b8+1","chain":null}
