{"id":"n-d1d6cf044938078a","revision":0,"first_index":0,"fingerprint":"f6-b0ee4e6a5f8b87bd","name":"theta_plus_one of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*6^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":[["64","192","108"],["512","2304","2592","864"],["4096","24576","41472","27648","6480"],["32768","245760","552960","552960","259200","46656"],["262144","2359296","6635520","8847360","6220800","2239488","326592"],["2097152","22020096","74317824","123863040","116121600","62705664","18289152","2239488"],["16777216","201326592","792723456","1585446912","1857945600","1337720832","585252864","143327232","15116544"],["134217728","1811939328","8153726976","19025362944","26754416640","24078974976","14046068736","5159780352","1088391168","100776960"],["1073741824","16106127360","81537269760","217432719360","356725555200","385263599616","280921374720","137594142720","43535646720","8062156800","665127936"],["8589934592","141733920768","797253304320","2391759912960","4484549836800","5650532794368","4944216195072","3027071139840","1277045637120","354734899200","58531258368","4353564672"],["68719476736","1236950581248","7653631721472","25512105738240","53814598041600","77493021179904","79107459121152","58119765884928","30649095290880","11351516774400","2809500401664","417942208512","28298170368"],["549755813888","10720238370816","72361609003008","265325899677696","621857577369600","1007409275338752","1175310821228544","1007409275338752","637501182050304","295139436134400","97396013924352","21732994842624","2943009718272","182849716224"],["4398046511104","92358976733184","675375017361408","2701500069445632","6964804866539520","12536648759771136","16454351497199616","16118548405420032","11900022064939008","6611123369410560","2727088389881856","811365140791296","164808544223232","20479168217088","1175462461440"],["35184372088832","791648371998720","6234230929489920","27015000694456320","75979689453158400","150439785117253632","219391353295994880","241778226081300480","204000378256097280","132222467388211200","65450121357164544","24340954223738880","6592341768929280","1228750093025280","141055495372800","7522959753216"],["281474976710656","6755399441055744","56998682783907840","265993852991569920","810450020833689600","1750572045000769536","2808209322188734464","3438623659822940160","3264006052097556480","2417782260813004800","1396269255619510272","623128428127715328","210954936605736960","52426670635745280","9027551703859200","962938848411648","47958868426752"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["364","6272","104272","1690304","26891200","421654016","6535637248","100353655808","1528917462016","23140372398080","348262604591104","5215839938527232","77784047778906112","1155648709858033664","17112490511359344640"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["64","192","108"]],"recurrence":{"denominator":{"coeffs":[["4","6"],["0","3"]]},"terms":[{"coeff":{"coeffs":[["32","96","54"],["0","24","18"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":192071,"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-bbe7a9ccf6944bc3"]},{"id":192072,"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-bbe7a9ccf6944bc3"]},{"id":192073,"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-bbe7a9ccf6944bc3"]},{"id":192074,"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-bbe7a9ccf6944bc3"]}],"region_exact":{},"incoming_edges":[{"id":"e-bbe7a9ccf6944bc3","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-d1d6cf044938078a","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":"7c0d72f3adbfb0a982b9fd77e812ab74caa45b7c548d349ab32864f437f2fb9e","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-c11cb6bca0e3e7dc","role":"sequence","name":"Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*6^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-053e33b9b4e87583","kind":"fingerprint_related","node_a_id":"n-2e272676adc6fe16","node_b_id":"n-d1d6cf044938078a","fingerprint":"f6-b0ee4e6a5f8b87bd","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b0ee4e6a5f8b87bd"},"other_id":"n-2e272676adc6fe16","other":{"id":"n-2e272676adc6fe16","name":"theta_plus_one of basis_map_A007318 of Triangle of coefficients in expansion of (1+3*x)^n.","first_index":0,"fingerprint":"f6-b0ee4e6a5f8b87bd","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","6"],["16","48","27"]],"card_rows":[["1"],["4","6"],["16","48","27"],["64","288","324","108"]],"row_sums_preview":["1","10","91","784","6517","52822","420175","3294172"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-6a5af5e1168bb679","steps":["basis A007318","\u03b8+1"],"edges":["e-5bade392cbb9f615","e-9ba16c70521df0ae"],"text":"A013610 \u2192 basis A007318 \u2192 \u03b8+1"},"headline":"A013610 \u2192 basis A007318 \u2192 \u03b8+1","chain":null}},{"id":"l-74306a1a736ab0d0","kind":"fingerprint_related","node_a_id":"n-d1d6cf044938078a","node_b_id":"n-6348aebb24bf7280","fingerprint":"f6-b0ee4e6a5f8b87bd","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b0ee4e6a5f8b87bd"},"other_id":"n-6348aebb24bf7280","other":{"id":"n-6348aebb24bf7280","name":"theta_plus_one of basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j.","first_index":0,"fingerprint":"f6-b0ee4e6a5f8b87bd","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["256","768","432"],["4096","18432","20736","6912"],["65536","393216","663552","442368","103680"]],"card_rows":[["256","768","432"],["4096","18432","20736","6912"],["65536","393216","663552","442368","103680"],["1048576","7864320","17694720","17694720","8294400","1492992"]],"row_sums_preview":["1456","50176","1668352","54089728","1721036800","53971714048","1673123135488","51381071773696"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a93042cd70c4761e","steps":["basis A013610","\u03b8+1"],"edges":["e-77aaa200c8a95097","e-0402be19fe211397"],"text":"A038330 \u2192 basis A013610 \u2192 \u03b8+1"},"headline":"A038330 \u2192 basis A013610 \u2192 \u03b8+1","chain":null}},{"id":"l-944d70083ae48cae","kind":"fingerprint_related","node_a_id":"n-bd4f560f4225207a","node_b_id":"n-d1d6cf044938078a","fingerprint":"f6-b0ee4e6a5f8b87bd","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b0ee4e6a5f8b87bd"},"other_id":"n-bd4f560f4225207a","other":{"id":"n-bd4f560f4225207a","name":"theta_plus_one of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*9^j.","first_index":0,"fingerprint":"f6-b0ee4e6a5f8b87bd","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["12","18"],["144","432","243"]],"card_rows":[["1"],["12","18"],["144","432","243"],["1728","7776","8748","2916"]],"row_sums_preview":["1","30","819","21168","527877","12835746","306307575","7204354164"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-f485cd65a8f8089f","steps":["\u03b8+1"],"edges":["e-4c0b69de31c9f368"],"text":"A038335 \u2192 \u03b8+1"},"headline":"A038335 \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:         64,          192,          108;\n  1:        512,         2304,         2592,           864;\n  2:       4096,        24576,        41472,         27648,          6480;\n  3:      32768,       245760,       552960,        552960,        259200,         46656;\n  4:     262144,      2359296,      6635520,       8847360,       6220800,       2239488,        326592;\n  5:    2097152,     22020096,     74317824,     123863040,     116121600,      62705664,      18289152,       2239488;\n  6:   16777216,    201326592,    792723456,    1585446912,    1857945600,    1337720832,     585252864,     143327232,      15116544;\n  7:  134217728,   1811939328,   8153726976,   19025362944,   26754416640,   24078974976,   14046068736,    5159780352,    1088391168,    100776960;\n  8: 1073741824,  16106127360,  81537269760,  217432719360,  356725555200,  385263599616,  280921374720,  137594142720,   43535646720,   8062156800,   665127936;\n  9: 8589934592, 141733920768, 797253304320, 2391759912960, 4484549836800, 5650532794368, 4944216195072, 3027071139840, 1277045637120, 354734899200, 58531258368, 4353564672;\n","recurrence_sentence":"The row polynomials satisfy (4+(3*n+6)*x)*P(n,x) = (32+(24*n+96)*x+(18*n+54)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 64 + 192*x + 108*x^2.","mathematica":"P[0, x_] := 64 + 192*x + 108*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((32 + (24*n + 96)*x + (18*n + 54)*x^2)*P[n - 1, x])/(4 + (3*n + 6)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 64 + 192*x + 108*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((32 + (24*n + 96)*x + (18*n + 54)*x^2)*v[i - 1])/(4 + (3*n + 6)*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-c11cb6bca0e3e7dc","steps":["\u03b8+1"],"edges":["e-bbe7a9ccf6944bc3"],"text":"A038284 \u2192 \u03b8+1"},"headline":"A038284 \u2192 \u03b8+1","chain":null}