{"id":"n-4b3fd66b22df45fb","revision":0,"first_index":0,"fingerprint":"f6-1e916cb7d7a3299c","name":"theta_plus_one_squared of derivative of Cube of lower triangular normalized binomial matrix.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[[],["1"],["6","8"],["27","72","27"],["108","432","324","64"],["405","2160","2430","960","125"],["1458","9720","14580","8640","2250","216"],["5103","40824","76545","60480","23625","4536","343"],["17496","163296","367416","362880","189000","54432","8232","512"],["59049","629856","1653372","1959552","1275750","489888","111132","13824","729"],["196830","2361960","7085880","9797760","7654500","3674160","1111320","207360","21870","1000"],["649539","8660520","29229255","46189440","42099750","24249456","9168390","2280960","360855","33000","1331"],["2125764","31177872","116917020","207852480","216513000","145496736","66012408","20528640","4330260","594000","47916","1728"],["6908733","110539728","455976378","900694080","1055500875","810624672","429080652","160123392","42220035","7722000","934362","67392","2197"],["22320522","386889048","1741000716","3782915136","4925670750","4255779528","2574483912","1120863744","354648294","81081000","13081068","1415232","92274","2744"]],"degrees":[-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["0","1","14","126","928","6080","36864","211456","1163264","6193152","32112640","162922496","811597824","3980394496","19260243968"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one_squared","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["1"]],"recurrence":{"denominator":{"coeffs":[["-9","12","-1"],["9","-21","3"],["0","9","-3"],["0","0","1"]]},"terms":[{"coeff":{"coeffs":[["0","0","0","0"],["27","0","-3","0"],["0","27","9","0"],["0","0","3","1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":176761,"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-b3efc1ba86f223a0"]},{"id":176762,"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-b3efc1ba86f223a0"]},{"id":176763,"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-b3efc1ba86f223a0"]}],"region_exact":{},"incoming_edges":[{"id":"e-b3efc1ba86f223a0","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-4b3fd66b22df45fb","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":"55155213b24c29308ce2c69c6f2af7c9c3e66b9da85de2c5001ba63194c01746","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-f5aba8978740eb12","role":"sequence","name":"derivative of Cube of lower triangular normalized binomial matrix.","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-479f1a0160a444c5","kind":"fingerprint_related","node_a_id":"n-fae37939907544d7","node_b_id":"n-4b3fd66b22df45fb","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-51a12c014558e493","other":{"id":"n-51a12c014558e493","name":"theta_plus_one_squared of basis_map_A013609 of Triangle whose (i,j)-th entry is 5^(i-j)*binomial(i,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"],["6","8"],["36","96","36"]],"card_rows":[["1"],["6","8"],["36","96","36"],["216","864","648","128"]],"row_sums_preview":["1","14","168","1856","19456","196608","1933312","18612224"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"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}},{"id":"l-74b291d9a2bd2555","kind":"fingerprint_related","node_a_id":"n-fd33170a2f06ac0d","node_b_id":"n-4b3fd66b22df45fb","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-7c5d74e88b78d46c","kind":"fingerprint_related","node_a_id":"n-4b3fd66b22df45fb","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-ef201465ca5f431e","kind":"fingerprint_related","node_a_id":"n-4b3fd66b22df45fb","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:     0;\n  1:     1;\n  2:     6,      8;\n  3:    27,     72,      27;\n  4:   108,    432,     324,      64;\n  5:   405,   2160,    2430,     960,     125;\n  6:  1458,   9720,   14580,    8640,    2250,    216;\n  7:  5103,  40824,   76545,   60480,   23625,   4536,    343;\n  8: 17496, 163296,  367416,  362880,  189000,  54432,   8232,   512;\n  9: 59049, 629856, 1653372, 1959552, 1275750, 489888, 111132, 13824, 729;\n","recurrence_sentence":"The row polynomials satisfy (9*n-9+(9*n^2-21*n+12)*x+(n^3-3*n^2+3*n-1)*x^2)*P(n,x) = (27*n+27*n^2*x+(3*n^3+9*n^2-3*n)*x^2+n^3*x^3)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((27*n + 27*n^2*x + (3*n^3 + 9*n^2 - 3*n)*x^2 + n^3*x^3)*P[n - 1, x])/(9*n - 9 + (9*n^2 - 21*n + 12)*x + (n^3 - 3*n^2 + 3*n - 1)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((27*n + 27*n^2*x + (3*n^3 + 9*n^2 - 3*n)*x^2 + n^3*x^3)*v[i - 1])/(9*n - 9 + (9*n^2 - 21*n + 12)*x + (n^3 - 3*n^2 + 3*n - 1)*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-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}
