{"id":"n-f7b66e37e15b5e24","revision":0,"first_index":0,"fingerprint":"f6-9e3a89ee8f65d0f5","name":"theta_plus_one_squared 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":false,"rows":[["144","288","81"],["1728","5184","2916","432"],["20736","82944","69984","20736","2025"],["248832","1244160","1399680","622080","121500","8748"],["2985984","17915904","25194240","14929920","4374000","629856","35721"],["35831808","250822656","423263232","313528320","122472000","26453952","3000564","139968"],["429981696","3439853568","6772211712","6019743744","2939328000","846526464","144027072","13436928","531441"],["5159780352","46438023168","104485552128","108355387392","63489484800","22856214528","5184974592","725594112","57395628","1968300"],["61917364224","619173642240","1567283281920","1857520926720","1269789696000","548549148672","155549237760","29023764480","3443737680","236196000","7144929"],["743008370688","8173092077568","22986821468160","30649095290880","23944605696000","12068081270784","4106499876864","957784227840","151524457920","15588936000","943130628","25509168"],["8916100448256","106993205379072","331010229141504","490385524654080","431002902528000","248257671856128","98555997044736","27584185761792","5454880485120","748268928000","67905405216","3673320192","89813529"],["106993205379072","1390911669927936","4694326886006784","7650014184603648","7470716977152000","4841024601194496","2196390791282688","717188829806592","170192271135744","29182488192000","3531081071232","286518974976","14010910524","312487308"],["1283918464548864","17974858503684096","65720576404094976","116836580273946624","125508045216153600","90365792555630592","46124206616936448","17212531915358208","4765383591800832","980531603251200","148305404991744","16045062598656","1176916484016","52497867744","1076168025"],["15407021574586368","231105323618795520","909977211749007360","1752548704109199360","2053768012627968000","1626584266001350656","922484132338728960","387281968095559680","122538435217735680","29415948097536000","5338994579702784","722027816939520","70614989040960","4724808096960","193710244500","3673320192"],["184884258895036416","2958148142320582656","12479687475414958080","25883796245305098240","32860288202047488000","28391289006569029632","17711695340903596032","8262015319371939840","2940922445225656320","806837433532416000","170847826550489088","27725868170477568","3389519473966080","302387718205440","18596183472000","705277476864","12440502369"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["513","10260","196425","3645000","66065625","1175512500","20605640625","356754375000","6112494140625","103797070312500","1748980634765625","29270773828125000","486938006103515625","8057247583007812500","132681848785400390625"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one_squared","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["144","288","81"]],"recurrence":{"denominator":{"coeffs":[["16","20","4"],["0","12","4"],["0","0","1"]]},"terms":[{"coeff":{"coeffs":[["192","432","204","27"],["0","144","108","18"],["0","0","12","3"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":192231,"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-1ffd4d6c3361e1c6"]},{"id":192232,"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-1ffd4d6c3361e1c6"]},{"id":192233,"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-1ffd4d6c3361e1c6"]},{"id":192234,"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-1ffd4d6c3361e1c6"]}],"region_exact":{},"incoming_edges":[{"id":"e-1ffd4d6c3361e1c6","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-f7b66e37e15b5e24","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":"80be2413ec489eaaf136321609289ee0972ddbf75f20cfaf0abf16d2c63e0b6c","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-405e23d49e27b447","kind":"fingerprint_related","node_a_id":"n-98ccf908b5b0e9d6","node_b_id":"n-f7b66e37e15b5e24","fingerprint":"f6-9e3a89ee8f65d0f5","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-9e3a89ee8f65d0f5"},"other_id":"n-98ccf908b5b0e9d6","other":{"id":"n-98ccf908b5b0e9d6","name":"theta_plus_one_squared of derivative of Triangle whose (i,j)-th entry is binomial(i,j)*4^(i-j).","first_index":0,"fingerprint":"f6-9e3a89ee8f65d0f5","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["1"],["8","8"]],"card_rows":[["1"],["8","8"],["48","96","27"],["256","768","432","64"]],"row_sums_preview":["0","1","16","171","1520","12125","90000","634375"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3dcaebc50e901d48","steps":["D","(\u03b8+1)\u00b2"],"edges":["e-62aaaa59f91d42c1","e-bbdc61da87e6d2cd"],"text":"A038231 \u2192 D \u2192 (\u03b8+1)\u00b2"},"headline":"A038231 \u2192 D \u2192 (\u03b8+1)\u00b2","chain":null}},{"id":"l-4621e9c6d3ee1d2d","kind":"fingerprint_related","node_a_id":"n-cb3b4ec925637cfb","node_b_id":"n-f7b66e37e15b5e24","fingerprint":"f6-9e3a89ee8f65d0f5","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-9e3a89ee8f65d0f5"},"other_id":"n-cb3b4ec925637cfb","other":{"id":"n-cb3b4ec925637cfb","name":"theta_plus_one_squared of basis_map_A013611 of Triangle whose (n,k)-th entry is 15^(n-k)*binomial(n,k).","first_index":0,"fingerprint":"f6-9e3a89ee8f65d0f5","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["16","16"],["256","512","144"]],"card_rows":[["1"],["16","16"],["256","512","144"],["4096","12288","6912","1024"]],"row_sums_preview":["1","32","912","24320","620800","15360000","371200000","8806400000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ff0214c333bfdcca","steps":["basis A013611","(\u03b8+1)\u00b2"],"edges":["e-9c08db68889738c4","e-ae46b299ff0b8e1b"],"text":"A027467 \u2192 basis A013611 \u2192 (\u03b8+1)\u00b2"},"headline":"A027467 \u2192 basis A013611 \u2192 (\u03b8+1)\u00b2","chain":null}},{"id":"l-69962333d0bdb814","kind":"fingerprint_related","node_a_id":"n-f7b66e37e15b5e24","node_b_id":"n-479804acd890a484","fingerprint":"f6-9e3a89ee8f65d0f5","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-9e3a89ee8f65d0f5"},"other_id":"n-479804acd890a484","other":{"id":"n-479804acd890a484","name":"theta_plus_one_squared of basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j.","first_index":0,"fingerprint":"f6-9e3a89ee8f65d0f5","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["256","512","144"],["4096","12288","6912","1024"],["65536","262144","221184","65536","6400"]],"card_rows":[["256","512","144"],["4096","12288","6912","1024"],["65536","262144","221184","65536","6400"],["1048576","5242880","5898240","2621440","512000","36864"]],"row_sums_preview":["912","24320","620800","15360000","371200000","8806400000","205824000000","4751360000000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a93042cd70c4761e","steps":["basis A007318","(\u03b8+1)\u00b2"],"edges":["e-7b68ca495e98f7ea","e-b7651ebc5ebdbb42"],"text":"A038330 \u2192 basis A007318 \u2192 (\u03b8+1)\u00b2"},"headline":"A038330 \u2192 basis A007318 \u2192 (\u03b8+1)\u00b2","chain":null}},{"id":"l-b81517f7b83c6635","kind":"fingerprint_related","node_a_id":"n-d861aaa140b3a14a","node_b_id":"n-f7b66e37e15b5e24","fingerprint":"f6-9e3a89ee8f65d0f5","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-9e3a89ee8f65d0f5"},"other_id":"n-d861aaa140b3a14a","other":{"id":"n-d861aaa140b3a14a","name":"theta_plus_one_squared of Triangle whose (i,j)-th entry is binomial(i,j)*4^(i-j).","first_index":0,"fingerprint":"f6-9e3a89ee8f65d0f5","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["4","4"],["16","32","9"]],"card_rows":[["1"],["4","4"],["16","32","9"],["64","192","108","16"]],"row_sums_preview":["1","8","57","380","2425","15000","90625","537500"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3dcaebc50e901d48","steps":["(\u03b8+1)\u00b2"],"edges":["e-831f84c152bb1a61"],"text":"A038231 \u2192 (\u03b8+1)\u00b2"},"headline":"A038231 \u2192 (\u03b8+1)\u00b2","chain":null}},{"id":"l-db0a669b2b81bd84","kind":"fingerprint_related","node_a_id":"n-2e63cfc7b87a7680","node_b_id":"n-f7b66e37e15b5e24","fingerprint":"f6-9e3a89ee8f65d0f5","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-9e3a89ee8f65d0f5"},"other_id":"n-2e63cfc7b87a7680","other":{"id":"n-2e63cfc7b87a7680","name":"theta_plus_one_squared of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j).","first_index":0,"fingerprint":"f6-9e3a89ee8f65d0f5","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["8","8"],["64","128","36"]],"card_rows":[["1"],["8","8"],["64","128","36"],["512","1536","864","128"]],"row_sums_preview":["1","16","228","3040","38800","480000","5800000","68800000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-f211cb6153c37c6d","steps":["basis A013609","(\u03b8+1)\u00b2"],"edges":["e-13f16c9d95bb60f6","e-50a6eeae9c184de2"],"text":"A027466 \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2"},"headline":"A027466 \u2192 basis A013609 \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:          144,           288,             81;\n  1:         1728,          5184,           2916,            432;\n  2:        20736,         82944,          69984,          20736,           2025;\n  3:       248832,       1244160,        1399680,         622080,         121500,           8748;\n  4:      2985984,      17915904,       25194240,       14929920,        4374000,         629856,         35721;\n  5:     35831808,     250822656,      423263232,      313528320,      122472000,       26453952,       3000564,       139968;\n  6:    429981696,    3439853568,     6772211712,     6019743744,     2939328000,      846526464,     144027072,     13436928,       531441;\n  7:   5159780352,   46438023168,   104485552128,   108355387392,    63489484800,    22856214528,    5184974592,    725594112,     57395628,     1968300;\n  8:  61917364224,  619173642240,  1567283281920,  1857520926720,  1269789696000,   548549148672,  155549237760,  29023764480,   3443737680,   236196000,   7144929;\n  9: 743008370688, 8173092077568, 22986821468160, 30649095290880, 23944605696000, 12068081270784, 4106499876864, 957784227840, 151524457920, 15588936000, 943130628, 25509168;\n","recurrence_sentence":"The row polynomials satisfy (16+(12*n+20)*x+(n^2+4*n+4)*x^2)*P(n,x) = (192+(144*n+432)*x+(12*n^2+108*n+204)*x^2+(3*n^2+18*n+27)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 144 + 288*x + 81*x^2.","mathematica":"P[0, x_] := 144 + 288*x + 81*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((192 + (144*n + 432)*x + (12*n^2 + 108*n + 204)*x^2 + (3*n^2 + 18*n + 27)*x^3)*P[n - 1, x])/(16 + (12*n + 20)*x + (n^2 + 4*n + 4)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 144 + 288*x + 81*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((192 + (144*n + 432)*x + (12*n^2 + 108*n + 204)*x^2 + (3*n^2 + 18*n + 27)*x^3)*v[i - 1])/(16 + (12*n + 20)*x + (n^2 + 4*n + 4)*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-615df6ea54d3bbef","steps":["(\u03b8+1)\u00b2"],"edges":["e-1ffd4d6c3361e1c6"],"text":"A038329 \u2192 (\u03b8+1)\u00b2"},"headline":"A038329 \u2192 (\u03b8+1)\u00b2","chain":null}
