{"id":"n-e782416b20ea362c","revision":0,"first_index":0,"fingerprint":"f6-7f34919e5b06c424","name":"theta_plus_one_squared of basis_map_A013611 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":[["225","1440","1296"],["3375","32400","58320","27648"],["50625","648000","1749600","1658880","518400"],["759375","12150000","43740000","62208000","38880000","8957952"],["11390625","218700000","984150000","1866240000","1749600000","806215680","146313216"],["170859375","3827250000","20667150000","48988800000","61236000000","42326323200","15362887680","2293235712"],["2562890625","65610000000","413343000000","1175731200000","1837080000000","1693052928000","921773260800","275188285440","34828517376"],["38443359375","1107168750000","7971615000000","26453952000000","49601160000000","57140536320000","41479796736000","18575209267200","4701849845760","515978035200"],["576650390625","18452812500000","149467781250000","566870400000000","1240029000000000","1714216089600000","1555492377600000","928760463360000","352638738432000","77396705280000","7492001071104"],["8649755859375","304471406250000","2740242656250000","11691702000000000","29229255000000000","47140942464000000","51331248460800000","38311369113600000","19395130613760000","6385228185600000","1236180176732160","106993205379072"],["129746337890625","4982259375000000","49324367812500000","233834040000000000","657658237500000000","1212195663360000000","1539937453824000000","1379209288089600000","872780877619200000","383113691136000000","111256215905894400","19258776968232960","1506820975755264"],["1946195068359375","80961714843750000","874386520312500000","4559763780000000000","14249261812500000000","29547269294400000000","42898257642240000000","44824301862912000000","34038454227148800000","18676792442880000000","7231654033883136000","1877730754402713600","293830090272276480","20970668254298112"],["29192926025390625","1307843085937500000","15301764105468750000","87050035800000000000","299234498062500000000","689436283536000000000","1126079263108800000000","1344729055887360000000","1191345897950208000000","784425282600960000000","379661836778864640000","131441152808189952000","30852159478589030400","4403840333402603520","288881654523494400"],["437893890380859375","21018906738281250000","264838224902343750000","1632188171250000000000","6120705642187500000000","15512316379560000000000","28151981577720000000000","37820504696832000000000","38293261005542400000000","29415948097536000000000","17084782655048908800000","7393564845460684800000","2313911960894177280000","495432037507792896000","64998372267786240000","3944197523094110208"],["6568408355712890625","336302507812500000000","4540083855468750000000","30132704700000000000000","122414112843750000000000","338450539190400000000000","675647557865280000000000","1008546791915520000000000","1148797830166272000000000","1008546791915520000000000","683391306201956352000000","354891112582112870400000","138834717653650636800000","39634563000623431680000","7799804672134348800000","946607405542586449920","53431550820665524224"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["2961","121743","4625505","166695327","5782609521","194872505967","6419170082241","207585709313535","6611393019483729","207881413038230607","6465187438904073249","199175581039117833567","6085586988116829360945","184589834664861349985583","5562970826435219854544769"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one_squared","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["225","1440","1296"]],"recurrence":{"denominator":{"coeffs":[["25","100","64"],["0","60","64"],["0","0","16"]]},"terms":[{"coeff":{"coeffs":[["375","2700","4080","1728"],["0","900","2160","1152"],["0","0","240","192"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":194067,"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-03698b142e87b517"]},{"id":194068,"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-03698b142e87b517"]},{"id":194069,"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-03698b142e87b517"]},{"id":194070,"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-03698b142e87b517"]}],"region_exact":{},"incoming_edges":[{"id":"e-03698b142e87b517","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-e782416b20ea362c","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":"d8529790660cec594cb153b301974d36195e3d1426a31ed2efa3eb762519954e","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-16c35c7477b7e922","role":"sequence","name":"basis_map_A013611 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-0811e9d756f5f785","kind":"fingerprint_related","node_a_id":"n-71be7c5970f096b1","node_b_id":"n-e782416b20ea362c","fingerprint":"f6-7f34919e5b06c424","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-7f34919e5b06c424"},"other_id":"n-71be7c5970f096b1","other":{"id":"n-71be7c5970f096b1","name":"theta_plus_one_squared of basis_map_A007318 of Triangle of coefficients in expansion of (1+4x)^n.","first_index":0,"fingerprint":"f6-7f34919e5b06c424","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["5","16"],["25","160","144"]],"card_rows":[["1"],["5","16"],["25","160","144"],["125","1200","2160","1024"]],"row_sums_preview":["1","21","329","4509","57105","685989","7932249","89104941"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-2e0409c0014ec240","steps":["basis A007318","(\u03b8+1)\u00b2"],"edges":["e-13c78408b476aecd","e-37d927a7a182dbd7"],"text":"A013611 \u2192 basis A007318 \u2192 (\u03b8+1)\u00b2"},"headline":"A013611 \u2192 basis A007318 \u2192 (\u03b8+1)\u00b2","chain":null}},{"id":"l-1ba6b9c69e31d815","kind":"fingerprint_related","node_a_id":"n-e782416b20ea362c","node_b_id":"n-d85cea0fb7505d23","fingerprint":"f6-7f34919e5b06c424","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-7f34919e5b06c424"},"other_id":"n-d85cea0fb7505d23","other":{"id":"n-d85cea0fb7505d23","name":"theta_plus_one_squared of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*8^j.","first_index":0,"fingerprint":"f6-7f34919e5b06c424","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["400","2560","2304"],["8000","76800","138240","65536"],["160000","2048000","5529600","5242880","1638400"]],"card_rows":[["400","2560","2304"],["8000","76800","138240","65536"],["160000","2048000","5529600","5242880","1638400"],["3200000","51200000","184320000","262144000","163840000","37748736"]],"row_sums_preview":["5264","288576","14618880","702452736","32490491904","1459895353344","64119300489216","2764687709306880"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-6874f381e9c7e0a2","steps":["basis A013609","(\u03b8+1)\u00b2"],"edges":["e-4eca5e8e987e069c","e-fc2dc75a36e0f895"],"text":"A038334 \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2"},"headline":"A038334 \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:           225,            1440,             1296;\n  1:          3375,           32400,            58320,             27648;\n  2:         50625,          648000,          1749600,           1658880,            518400;\n  3:        759375,        12150000,         43740000,          62208000,          38880000,           8957952;\n  4:      11390625,       218700000,        984150000,        1866240000,        1749600000,         806215680,         146313216;\n  5:     170859375,      3827250000,      20667150000,       48988800000,       61236000000,       42326323200,       15362887680,        2293235712;\n  6:    2562890625,     65610000000,     413343000000,     1175731200000,     1837080000000,     1693052928000,      921773260800,      275188285440,       34828517376;\n  7:   38443359375,   1107168750000,    7971615000000,    26453952000000,    49601160000000,    57140536320000,    41479796736000,    18575209267200,     4701849845760,     515978035200;\n  8:  576650390625,  18452812500000,  149467781250000,   566870400000000,  1240029000000000,  1714216089600000,  1555492377600000,   928760463360000,   352638738432000,   77396705280000,    7492001071104;\n  9: 8649755859375, 304471406250000, 2740242656250000, 11691702000000000, 29229255000000000, 47140942464000000, 51331248460800000, 38311369113600000, 19395130613760000, 6385228185600000, 1236180176732160, 106993205379072;\n","recurrence_sentence":"The row polynomials satisfy (25+(60*n+100)*x+(16*n^2+64*n+64)*x^2)*P(n,x) = (375+(900*n+2700)*x+(240*n^2+2160*n+4080)*x^2+(192*n^2+1152*n+1728)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 225 + 1440*x + 1296*x^2.","mathematica":"P[0, x_] := 225 + 1440*x + 1296*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((375 + (900*n + 2700)*x + (240*n^2 + 2160*n + 4080)*x^2 + (192*n^2 + 1152*n + 1728)*x^3)*P[n - 1, x])/(25 + (60*n + 100)*x + (16*n^2 + 64*n + 64)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 225 + 1440*x + 1296*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((375 + (900*n + 2700)*x + (240*n^2 + 2160*n + 4080)*x^2 + (192*n^2 + 1152*n + 1728)*x^3)*v[i - 1])/(25 + (60*n + 100)*x + (16*n^2 + 64*n + 64)*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":["basis A013611","(\u03b8+1)\u00b2"],"edges":["e-a5d7492b5780a0c3","e-03698b142e87b517"],"text":"A038329 \u2192 basis A013611 \u2192 (\u03b8+1)\u00b2"},"headline":"A038329 \u2192 basis A013611 \u2192 (\u03b8+1)\u00b2","chain":null}