{"id":"n-51a12c014558e493","revision":0,"first_index":0,"fingerprint":"f6-1e916cb7d7a3299c","name":"theta_plus_one_squared of basis_map_A013609 of Triangle whose (i,j)-th entry is 5^(i-j)*binomial(i,j).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["6","8"],["36","96","36"],["216","864","648","128"],["1296","6912","7776","3072","400"],["7776","51840","77760","46080","12000","1152"],["46656","373248","699840","552960","216000","41472","3136"],["279936","2612736","5878656","5806080","3024000","870912","131712","8192"],["1679616","17915904","47029248","55738368","36288000","13934592","3161088","393216","20736"],["10077696","120932352","362797056","501645312","391910400","188116992","56899584","10616832","1119744","51200"],["60466176","806215680","2720977920","4299816960","3919104000","2257403904","853493760","212336640","33592320","3072000","123904"],["362797056","5321023488","19953838080","35473489920","36951552000","24831442944","11266117632","3503554560","739031040","101376000","8177664","294912"],["2176782336","34828517376","143667634176","283787919360","332563968000","255409127424","135193411584","50451185664","13302558720","2433024000","294395904","21233664","692224"],["13060694016","226385362944","1018734133248","2213545771008","2882221056000","2490238992384","1506440871936","655865413632","207519916032","47443968000","7654293504","828112896","53993472","1605632"],["78364164096","1462797729792","7131138932736","16903440433152","24210656870400","23242230595584","15817629155328","7870384963584","2905278824448","797058662400","160740163584","23187161088","2267725824","134873088","3686400"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","14","168","1856","19456","196608","1933312","18612224","176160768","1644167168","15166603264","138512695296","1254130450432","11269994184704","100605313941504"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one_squared","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["9","-3","0"],["0","9","0"],["0","0","1"]]},"terms":[{"coeff":{"coeffs":[["54","54","18","2"],["0","54","30","4"],["0","0","6","2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":183857,"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-c498116e516908fe"]},{"id":183858,"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-c498116e516908fe"]},{"id":183859,"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-c498116e516908fe"]},{"id":183860,"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-c498116e516908fe"]}],"region_exact":{},"incoming_edges":[{"id":"e-c498116e516908fe","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-51a12c014558e493","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":"7551f9f9b52b3d811ac457eebe39fa863906b019eff5235a47a28e64f250fe3e","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-ff3307aeb79ab12b","role":"sequence","name":"basis_map_A013609 of Triangle whose (i,j)-th entry is 5^(i-j)*binomial(i,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-0355b3de0d3e191c","kind":"fingerprint_related","node_a_id":"n-fae37939907544d7","node_b_id":"n-51a12c014558e493","fingerprint":"f6-1e916cb7d7a3299c","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1e916cb7d7a3299c"},"other_id":"n-fae37939907544d7","other":{"id":"n-fae37939907544d7","name":"theta_plus_one_squared of Cube of lower triangular normalized binomial matrix.","first_index":0,"fingerprint":"f6-1e916cb7d7a3299c","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["3","4"],["9","24","9"]],"card_rows":[["1"],["3","4"],["9","24","9"],["27","108","81","16"]],"row_sums_preview":["1","7","42","232","1216","6144","30208","145408"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-8d887bfc40d036b9","steps":["(\u03b8+1)\u00b2"],"edges":["e-32b8bc759b20a254"],"text":"A027465 \u2192 (\u03b8+1)\u00b2"},"headline":"A027465 \u2192 (\u03b8+1)\u00b2","chain":null}},{"id":"l-70c1568d7e63f58d","kind":"fingerprint_related","node_a_id":"n-4b3fd66b22df45fb","node_b_id":"n-51a12c014558e493","fingerprint":"f6-1e916cb7d7a3299c","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1e916cb7d7a3299c"},"other_id":"n-4b3fd66b22df45fb","other":{"id":"n-4b3fd66b22df45fb","name":"theta_plus_one_squared of derivative of Cube of lower triangular normalized binomial matrix.","first_index":0,"fingerprint":"f6-1e916cb7d7a3299c","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["1"],["6","8"]],"card_rows":[["1"],["6","8"],["27","72","27"],["108","432","324","64"]],"row_sums_preview":["0","1","14","126","928","6080","36864","211456"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-8d887bfc40d036b9","steps":["D","(\u03b8+1)\u00b2"],"edges":["e-6602f7f73d905d9e","e-b3efc1ba86f223a0"],"text":"A027465 \u2192 D \u2192 (\u03b8+1)\u00b2"},"headline":"A027465 \u2192 D \u2192 (\u03b8+1)\u00b2","chain":null}},{"id":"l-8bc6f4d7ea10e90c","kind":"fingerprint_related","node_a_id":"n-fd33170a2f06ac0d","node_b_id":"n-51a12c014558e493","fingerprint":"f6-1e916cb7d7a3299c","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1e916cb7d7a3299c"},"other_id":"n-fd33170a2f06ac0d","other":{"id":"n-fd33170a2f06ac0d","name":"theta_plus_one_squared of basis_map_A007318 of Triangle read by rows: T(n,k) is the number of ternary words of length n having k levels (n >= 1, 0 <= k <= n-1). A level is a pair of identical consecutive letters.","first_index":0,"fingerprint":"f6-1e916cb7d7a3299c","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["3"],["9","12"],["27","72","27"]],"card_rows":[["3"],["9","12"],["27","72","27"],["81","324","243","48"]],"row_sums_preview":["3","21","126","696","3648","18432","90624","436224"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-51b4f5e7901c2b50","steps":["basis A007318","(\u03b8+1)\u00b2"],"edges":["e-18f137fab6db112d","e-7521e5dad2079159"],"text":"A120910 \u2192 basis A007318 \u2192 (\u03b8+1)\u00b2"},"headline":"A120910 \u2192 basis A007318 \u2192 (\u03b8+1)\u00b2","chain":null}},{"id":"l-c4f269ce56d2adf3","kind":"fingerprint_related","node_a_id":"n-51a12c014558e493","node_b_id":"n-70ef400185a6f340","fingerprint":"f6-1e916cb7d7a3299c","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1e916cb7d7a3299c"},"other_id":"n-70ef400185a6f340","other":{"id":"n-70ef400185a6f340","name":"theta_plus_one_squared of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j.","first_index":0,"fingerprint":"f6-1e916cb7d7a3299c","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["144","384","144"],["1728","6912","5184","1024"],["20736","110592","124416","49152","6400"]],"card_rows":[["144","384","144"],["1728","6912","5184","1024"],["20736","110592","124416","49152","6400"],["248832","1658880","2488320","1474560","384000","36864"]],"row_sums_preview":["672","14848","311296","6291456","123731968","2382364672","45097156608","841813590016"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a93042cd70c4761e","steps":["(\u03b8+1)\u00b2"],"edges":["e-1cde5008aff691e0"],"text":"A038330 \u2192 (\u03b8+1)\u00b2"},"headline":"A038330 \u2192 (\u03b8+1)\u00b2","chain":null}},{"id":"l-d9fe2ff37322a33d","kind":"fingerprint_related","node_a_id":"n-51a12c014558e493","node_b_id":"n-ef7c66a14cbd29c0","fingerprint":"f6-1e916cb7d7a3299c","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1e916cb7d7a3299c"},"other_id":"n-ef7c66a14cbd29c0","other":{"id":"n-ef7c66a14cbd29c0","name":"theta_plus_one_squared of basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*1^j.","first_index":0,"fingerprint":"f6-1e916cb7d7a3299c","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["9","12"],["81","216","81"]],"card_rows":[["1"],["9","12"],["81","216","81"],["729","2916","2187","432"]],"row_sums_preview":["1","21","378","6264","98496","1492992","22021632","318007296"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-366b2525683554ec","steps":["basis A013610","(\u03b8+1)\u00b2"],"edges":["e-b4f43a8046f9cfa3","e-1b8f2794be6fba2c"],"text":"A038279 \u2192 basis A013610 \u2192 (\u03b8+1)\u00b2"},"headline":"A038279 \u2192 basis A013610 \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:        1;\n  1:        6,         8;\n  2:       36,        96,        36;\n  3:      216,       864,       648,       128;\n  4:     1296,      6912,      7776,      3072,       400;\n  5:     7776,     51840,     77760,     46080,     12000,      1152;\n  6:    46656,    373248,    699840,    552960,    216000,     41472,     3136;\n  7:   279936,   2612736,   5878656,   5806080,   3024000,    870912,   131712,     8192;\n  8:  1679616,  17915904,  47029248,  55738368,  36288000,  13934592,  3161088,   393216,   20736;\n  9: 10077696, 120932352, 362797056, 501645312, 391910400, 188116992, 56899584, 10616832, 1119744, 51200;\n","recurrence_sentence":"The row polynomials satisfy (9+(9*n-3)*x+n^2*x^2)*P(n,x) = (54+(54*n+54)*x+(6*n^2+30*n+18)*x^2+(2*n^2+4*n+2)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((54 + (54*n + 54)*x + (6*n^2 + 30*n + 18)*x^2 + (2*n^2 + 4*n + 2)*x^3)*P[n - 1, x])/(9 + (9*n - 3)*x + n^2*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((54 + (54*n + 54)*x + (6*n^2 + 30*n + 18)*x^2 + (2*n^2 + 4*n + 2)*x^3)*v[i - 1])/(9 + (9*n - 3)*x + n^2*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-7b91a7a037fb3513","steps":["basis A013609","(\u03b8+1)\u00b2"],"edges":["e-037f2552a98ec0cc","e-c498116e516908fe"],"text":"A038243 \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2"},"headline":"A038243 \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2","chain":null}