{"id":"n-07cb3de7e9993649","revision":0,"first_index":0,"fingerprint":"f6-af50e1602aec03e4","name":"theta_plus_one 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":[["144","144","27"],["1728","2592","972","108"],["20736","41472","23328","5184","405"],["248832","622080","466560","155520","24300","1458"],["2985984","8957952","8398080","3732480","874800","104976","5103"],["35831808","125411328","141087744","78382080","24494400","4408992","428652","17496"],["429981696","1719926784","2257403904","1504935936","587865600","141087744","20575296","1679616","59049"],["5159780352","23219011584","34828517376","27088846848","12697896960","3809369088","740710656","90699264","6377292","196830"],["61917364224","309586821120","522427760640","464380231680","253957939200","91424858112","22221319680","3627970560","382637520","23619600","649539"],["743008370688","4086546038784","7662273822720","7662273822720","4788921139200","2011346878464","586642839552","119723028480","16836050880","1558893600","85739148","2125764"],["8916100448256","53496602689536","110336743047168","122596381163520","86200580505600","41376278642688","14079428149248","3448023220224","606097831680","74826892800","6173218656","306110016","6908733"],["106993205379072","695455834963968","1564775628668928","1912503546150912","1494143395430400","806837433532416","313770113040384","89648603725824","18910252348416","2918248819200","321007370112","23876581248","1077762348","22320522"],["1283918464548864","8987429251842048","21906858801364992","29209145068486656","25101609043230720","15060965425938432","6589172373848064","2151566489419776","529487065755648","98053160325120","13482309544704","1337088549888","90532037232","3749847696","71744535"],["15407021574586368","115552661809397760","303325737249669120","438137176027299840","410753602525593600","271097377666891776","131783447476961280","48410246011944960","13615381690859520","2941594809753600","485363143609344","60168984744960","5431922233920","337486292640","12914016300","229582512"],["184884258895036416","1479074071160291328","4159895825138319360","6470949061326274560","6572057640409497600","4731881501094838272","2530242191557656576","1032751914921492480","326769160580628480","80683743353241600","15531620595499008","2310489014206464","260732267228160","21599122728960","1239745564800","44079842304","731794257"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["315","5400","91125","1518750","25059375","410062500","6663515625","107641406250","1729951171875","27679218750000","441137548828125","7006302246093750","110933118896484375","1751575561523437500","27587315093994140625"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["144","144","27"]],"recurrence":{"denominator":{"coeffs":[["4","2"],["0","1"]]},"terms":[{"coeff":{"coeffs":[["48","48","9"],["0","12","3"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":192227,"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-55c0a4c5884c9aec"]},{"id":192228,"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-55c0a4c5884c9aec"]},{"id":192229,"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-55c0a4c5884c9aec"]},{"id":192230,"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-55c0a4c5884c9aec"]}],"region_exact":{},"incoming_edges":[{"id":"e-55c0a4c5884c9aec","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-07cb3de7e9993649","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":"d06802e811ff98ff6c90b14288d22be84c87a978fb70f8c01e6adf56d2cfbabf","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-615df6ea54d3bbef","role":"sequence","name":"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-184e34700c60a7e9","kind":"fingerprint_related","node_a_id":"n-7af05cad24952808","node_b_id":"n-07cb3de7e9993649","fingerprint":"f6-af50e1602aec03e4","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-af50e1602aec03e4"},"other_id":"n-7af05cad24952808","other":{"id":"n-7af05cad24952808","name":"theta_plus_one of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j).","first_index":0,"fingerprint":"f6-af50e1602aec03e4","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["8","4"],["64","64","12"]],"card_rows":[["1"],["8","4"],["64","64","12"],["512","768","288","32"]],"row_sums_preview":["1","12","140","1600","18000","200000","2200000","24000000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-f211cb6153c37c6d","steps":["basis A013609","\u03b8+1"],"edges":["e-13f16c9d95bb60f6","e-85b1ce05a797317c"],"text":"A027466 \u2192 basis A013609 \u2192 \u03b8+1"},"headline":"A027466 \u2192 basis A013609 \u2192 \u03b8+1","chain":null}},{"id":"l-213bc1c46175007c","kind":"fingerprint_related","node_a_id":"n-07cb3de7e9993649","node_b_id":"n-defea092f45b33f9","fingerprint":"f6-af50e1602aec03e4","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-af50e1602aec03e4"},"other_id":"n-defea092f45b33f9","other":{"id":"n-defea092f45b33f9","name":"theta_plus_one of basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j.","first_index":0,"fingerprint":"f6-af50e1602aec03e4","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["256","256","48"],["4096","6144","2304","256"],["65536","131072","73728","16384","1280"]],"card_rows":[["256","256","48"],["4096","6144","2304","256"],["65536","131072","73728","16384","1280"],["1048576","2621440","1966080","655360","102400","6144"]],"row_sums_preview":["560","12800","288000","6400000","140800000","3072000000","66560000000","1433600000000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a93042cd70c4761e","steps":["basis A007318","\u03b8+1"],"edges":["e-7b68ca495e98f7ea","e-8078ea16fa547745"],"text":"A038330 \u2192 basis A007318 \u2192 \u03b8+1"},"headline":"A038330 \u2192 basis A007318 \u2192 \u03b8+1","chain":null}},{"id":"l-3a400b9f2e43f748","kind":"fingerprint_related","node_a_id":"n-033b1e898256e58f","node_b_id":"n-07cb3de7e9993649","fingerprint":"f6-af50e1602aec03e4","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-af50e1602aec03e4"},"other_id":"n-033b1e898256e58f","other":{"id":"n-033b1e898256e58f","name":"theta_plus_one of derivative of Triangle whose (i,j)-th entry is binomial(i,j)*4^(i-j).","first_index":0,"fingerprint":"f6-af50e1602aec03e4","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["1"],["8","4"]],"card_rows":[["1"],["8","4"],["48","48","9"],["256","384","144","16"]],"row_sums_preview":["0","1","12","105","800","5625","37500","240625"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3dcaebc50e901d48","steps":["D","\u03b8+1"],"edges":["e-62aaaa59f91d42c1","e-214d5046e1129b44"],"text":"A038231 \u2192 D \u2192 \u03b8+1"},"headline":"A038231 \u2192 D \u2192 \u03b8+1","chain":null}},{"id":"l-41a580c04d487805","kind":"fingerprint_related","node_a_id":"n-07cb3de7e9993649","node_b_id":"n-7ec7c0d60a887c3d","fingerprint":"f6-af50e1602aec03e4","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-af50e1602aec03e4"},"other_id":"n-7ec7c0d60a887c3d","other":{"id":"n-7ec7c0d60a887c3d","name":"basis_map_A013611 of one_plus_derivative of 3n + 1 preceded by n zeros.","first_index":0,"fingerprint":"f6-af50e1602aec03e4","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["8","16"],["21","112","112"]],"card_rows":[["1"],["8","16"],["21","112","112"],["40","360","960","640"]],"row_sums_preview":["1","24","245","2000","14625","100000","653125","4125000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ccc86c3dcf2ac685","steps":["D+1","basis A013611"],"edges":["e-133ff869421f85a2","e-96965e8c131c21aa"],"text":"A131431 \u2192 D+1 \u2192 basis A013611"},"headline":"A131431 \u2192 D+1 \u2192 basis A013611","chain":null}},{"id":"l-9fb0355e83bbb20d","kind":"fingerprint_related","node_a_id":"n-07cb3de7e9993649","node_b_id":"n-8ea4f8a23684cfb4","fingerprint":"f6-af50e1602aec03e4","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-af50e1602aec03e4"},"other_id":"n-8ea4f8a23684cfb4","other":{"id":"n-8ea4f8a23684cfb4","name":"basis_map_A013611 of one_plus_derivative of Infinitesimal Pascal matrix: generator (lower triangular matrix representation) of the Pascal matrix, the classical operator xDx, iterated Laguerre transforms, associated matrices of the list partition transform and general Euler transformation for sequences.","first_index":0,"fingerprint":"f6-af50e1602aec03e4","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["1"],["4","8"]],"card_rows":[["1"],["4","8"],["9","48","48"],["16","144","384","256"]],"row_sums_preview":["0","1","12","105","800","5625","37500","240625"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-1c11ffc3c4cd47ed","steps":["D+1","basis A013611"],"edges":["e-9e777fb0c714fbf5","e-53401cb7af1f1125"],"text":"A132440 \u2192 D+1 \u2192 basis A013611"},"headline":"A132440 \u2192 D+1 \u2192 basis A013611","chain":null}},{"id":"l-a20906be8f36d184","kind":"fingerprint_related","node_a_id":"n-edc7bfe25fbf7be9","node_b_id":"n-07cb3de7e9993649","fingerprint":"f6-af50e1602aec03e4","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-af50e1602aec03e4"},"other_id":"n-edc7bfe25fbf7be9","other":{"id":"n-edc7bfe25fbf7be9","name":"theta_plus_one of Triangle whose (i,j)-th entry is binomial(i,j)*4^(i-j).","first_index":0,"fingerprint":"f6-af50e1602aec03e4","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","2"],["16","16","3"]],"card_rows":[["1"],["4","2"],["16","16","3"],["64","96","36","4"]],"row_sums_preview":["1","6","35","200","1125","6250","34375","187500"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3dcaebc50e901d48","steps":["\u03b8+1"],"edges":["e-cc808b1e75dd0280"],"text":"A038231 \u2192 \u03b8+1"},"headline":"A038231 \u2192 \u03b8+1","chain":null}},{"id":"l-c8ca78f9e027106c","kind":"fingerprint_related","node_a_id":"n-f5b145ff37c5f0de","node_b_id":"n-07cb3de7e9993649","fingerprint":"f6-af50e1602aec03e4","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-af50e1602aec03e4"},"other_id":"n-f5b145ff37c5f0de","other":{"id":"n-f5b145ff37c5f0de","name":"theta_plus_one of basis_map_A013611 of Triangle whose (n,k)-th entry is 15^(n-k)*binomial(n,k).","first_index":0,"fingerprint":"f6-af50e1602aec03e4","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["16","8"],["256","256","48"]],"card_rows":[["1"],["16","8"],["256","256","48"],["4096","6144","2304","256"]],"row_sums_preview":["1","24","560","12800","288000","6400000","140800000","3072000000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ff0214c333bfdcca","steps":["basis A013611","\u03b8+1"],"edges":["e-9c08db68889738c4","e-1de2e8dd43c7bdab"],"text":"A027467 \u2192 basis A013611 \u2192 \u03b8+1"},"headline":"A027467 \u2192 basis A013611 \u2192 \u03b8+1","chain":null}},{"id":"l-d5c367aa8c8345ca","kind":"fingerprint_related","node_a_id":"n-07cb3de7e9993649","node_b_id":"n-fda646f62e4dccbb","fingerprint":"f6-af50e1602aec03e4","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-af50e1602aec03e4"},"other_id":"n-fda646f62e4dccbb","other":{"id":"n-fda646f62e4dccbb","name":"basis_map_A013611 of one_plus_derivative of Triangle read by rows: n-th row is n^2 preceded by (n-1) zeros.","first_index":0,"fingerprint":"f6-af50e1602aec03e4","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["8","16"],["27","144","144"]],"card_rows":[["1"],["8","16"],["27","144","144"],["64","576","1536","1024"]],"row_sums_preview":["1","24","315","3200","28125","225000","1684375","12000000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7b3743413970d91c","steps":["\u03b8+1","D+1","basis A013611"],"edges":["e-06738ea5bb413735","e-2dc8b6d950a061ae","e-84a653b2d8905f85"],"text":"image under \u03b8+1 \u2192 \u03b8+1 \u2192 D+1 \u2192 basis A013611"},"headline":"image under \u03b8+1 \u2192 \u03b8+1 \u2192 D+1 \u2192 basis A013611","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:          144,           144,            27;\n  1:         1728,          2592,           972,           108;\n  2:        20736,         41472,         23328,          5184,           405;\n  3:       248832,        622080,        466560,        155520,         24300,          1458;\n  4:      2985984,       8957952,       8398080,       3732480,        874800,        104976,         5103;\n  5:     35831808,     125411328,     141087744,      78382080,      24494400,       4408992,       428652,        17496;\n  6:    429981696,    1719926784,    2257403904,    1504935936,     587865600,     141087744,     20575296,      1679616,       59049;\n  7:   5159780352,   23219011584,   34828517376,   27088846848,   12697896960,    3809369088,    740710656,     90699264,     6377292,     196830;\n  8:  61917364224,  309586821120,  522427760640,  464380231680,  253957939200,   91424858112,  22221319680,   3627970560,   382637520,   23619600,   649539;\n  9: 743008370688, 4086546038784, 7662273822720, 7662273822720, 4788921139200, 2011346878464, 586642839552, 119723028480, 16836050880, 1558893600, 85739148, 2125764;\n","recurrence_sentence":"The row polynomials satisfy (4+(n+2)*x)*P(n,x) = (48+(12*n+48)*x+(3*n+9)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 144 + 144*x + 27*x^2.","mathematica":"P[0, x_] := 144 + 144*x + 27*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((48 + (12*n + 48)*x + (3*n + 9)*x^2)*P[n - 1, x])/(4 + (n + 2)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 144 + 144*x + 27*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((48 + (12*n + 48)*x + (3*n + 9)*x^2)*v[i - 1])/(4 + (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":["\u03b8+1"],"edges":["e-55c0a4c5884c9aec"],"text":"A038329 \u2192 \u03b8+1"},"headline":"A038329 \u2192 \u03b8+1","chain":null}
