{"id":"n-e226c8909291af75","revision":0,"first_index":0,"fingerprint":"f6-476cf4c2bd2dc965","name":"theta_plus_one_squared of Triangle read by rows: (i,j)-th entry is binomial(i,j)*3^(i-j)*8^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":false,"rows":[["9","192","576"],["27","864","5184","8192"],["81","3456","31104","98304","102400"],["243","12960","155520","737280","1536000","1179648"],["729","46656","699840","4423680","13824000","21233664","12845056"],["2187","163296","2939328","23224320","96768000","222953472","269746176","134217728"],["6561","559872","11757312","111476736","580608000","1783627776","3236954112","3221225472","1358954496"],["19683","1889568","45349632","501645312","3135283200","12039487488","29132587008","43486543872","36691771392","13421772800"],["59049","6298560","170061120","2149908480","15676416000","72236924928","218494402560","434865438720","550376570880","402653184000","129922760704"],["177147","20785248","623557440","8868372480","73903104000","397303087104","1442063056896","3587639869440","6054142279680","6643777536000","4287451103232","1236950581248"],["531441","68024448","2244806784","35473489920","332563968000","2043273019392","8652378341376","25831007059968","54487280517120","79725330432000","77174119858176","44530220924928","11613591568384"],["1594323","221079456","7958860416","138346610688","1441110528000","9960955969536","48206107901952","167901545889792","425000788033536","777321971712000","1003263558156288","868339308036096","452930071166976","107752139522048"],["4782969","714256704","27856011456","528232513536","6052664217600","46484461191168","253082066485248","1007409275338752","2975005516234752","6529504562380800","10534267360641024","12156750312505344","9511531494506496","4525589859926016","989560464998400"],["14348907","2295825120","96424655040","1980871925760","24760899072000","209180075360256","1265410332426240","5666677173780480","19125035461509120","48971284217856000","94808406245769216","136763441015685120","142672972417597440","101825771848335360","44530220924928000","9007199254740992"],["43046721","7346640384","330598817280","7313988648960","99043596288000","912785783390208","6073969595645952","30222278260162560","114750212769054720","335803091779584000","758467249966153728","1312929033750577152","1712075669011169280","1629212349573365760","1068725302198272000","432345564227567616","81346268269379584"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["777","14267","235345","3621651","53073625","750014507","10305170337","138456349955","1826552025001","23732743509915","304427552541937","3862264085061107","48535794845990265","604872439473815051","7482970470757763905"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one_squared","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["9","192","576"]],"recurrence":{"denominator":{"coeffs":[["9","120","256"],["0","72","256"],["0","0","64"]]},"terms":[{"coeff":{"coeffs":[["27","648","3264","4608"],["0","216","1728","3072"],["0","0","192","512"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":190280,"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-4f986067612de658"]},{"id":190281,"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-4f986067612de658"]},{"id":190282,"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-4f986067612de658"]},{"id":190283,"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-4f986067612de658"]}],"region_exact":{},"incoming_edges":[{"id":"e-4f986067612de658","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-e226c8909291af75","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":"40c520d2b4a492202f2fa6e0a430ffc64a5bb75974443f5a342f7e89c1c3a2d9","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-26633b980c3508e8","role":"sequence","name":"Triangle read by rows: (i,j)-th entry is binomial(i,j)*3^(i-j)*8^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-a4734e1bf2279fd1","kind":"fingerprint_related","node_a_id":"n-6ba6b6f4df801f31","node_b_id":"n-e226c8909291af75","fingerprint":"f6-476cf4c2bd2dc965","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-476cf4c2bd2dc965"},"other_id":"n-6ba6b6f4df801f31","other":{"id":"n-6ba6b6f4df801f31","name":"theta_plus_one_squared of basis_map_A013611 of Triangle of coefficients in expansion of (1+2*x)^n.","first_index":0,"fingerprint":"f6-476cf4c2bd2dc965","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["3","32"],["9","192","576"]],"card_rows":[["1"],["3","32"],["9","192","576"],["27","864","5184","8192"]],"row_sums_preview":["1","35","777","14267","235345","3621651","53073625","750014507"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c44f7114310405ad","steps":["x \u21a6 x/(x+1)","(\u03b8+1)\u00b2"],"edges":["e-22d203db774fd047","e-cd84c5049baa5cd5"],"text":"A013622 \u2192 x \u21a6 x/(x+1) \u2192 (\u03b8+1)\u00b2"},"headline":"A013622 \u2192 x \u21a6 x/(x+1) \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:      9,      192,       576;\n  1:     27,      864,      5184,       8192;\n  2:     81,     3456,     31104,      98304,      102400;\n  3:    243,    12960,    155520,     737280,     1536000,      1179648;\n  4:    729,    46656,    699840,    4423680,    13824000,     21233664,      12845056;\n  5:   2187,   163296,   2939328,   23224320,    96768000,    222953472,     269746176,     134217728;\n  6:   6561,   559872,  11757312,  111476736,   580608000,   1783627776,    3236954112,    3221225472,    1358954496;\n  7:  19683,  1889568,  45349632,  501645312,  3135283200,  12039487488,   29132587008,   43486543872,   36691771392,   13421772800;\n  8:  59049,  6298560, 170061120, 2149908480, 15676416000,  72236924928,  218494402560,  434865438720,  550376570880,  402653184000,  129922760704;\n  9: 177147, 20785248, 623557440, 8868372480, 73903104000, 397303087104, 1442063056896, 3587639869440, 6054142279680, 6643777536000, 4287451103232, 1236950581248;\n","recurrence_sentence":"The row polynomials satisfy (9+(72*n+120)*x+(64*n^2+256*n+256)*x^2)*P(n,x) = (27+(216*n+648)*x+(192*n^2+1728*n+3264)*x^2+(512*n^2+3072*n+4608)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 9 + 192*x + 576*x^2.","mathematica":"P[0, x_] := 9 + 192*x + 576*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((27 + (216*n + 648)*x + (192*n^2 + 1728*n + 3264)*x^2 + (512*n^2 + 3072*n + 4608)*x^3)*P[n - 1, x])/(9 + (72*n + 120)*x + (64*n^2 + 256*n + 256)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 9 + 192*x + 576*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((27 + (216*n + 648)*x + (192*n^2 + 1728*n + 3264)*x^2 + (512*n^2 + 3072*n + 4608)*x^3)*v[i - 1])/(9 + (72*n + 120)*x + (64*n^2 + 256*n + 256)*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-26633b980c3508e8","steps":["(\u03b8+1)\u00b2"],"edges":["e-4f986067612de658"],"text":"A038226 \u2192 (\u03b8+1)\u00b2"},"headline":"A038226 \u2192 (\u03b8+1)\u00b2","chain":null}