{"id":"n-033b1e898256e58f","revision":0,"first_index":0,"fingerprint":"f6-af50e1602aec03e4","name":"theta_plus_one of derivative of Triangle whose (i,j)-th entry is binomial(i,j)*4^(i-j).","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":true,"rows":[[],["1"],["8","4"],["48","48","9"],["256","384","144","16"],["1280","2560","1440","320","25"],["6144","15360","11520","3840","600","36"],["28672","86016","80640","35840","8400","1008","49"],["131072","458752","516096","286720","89600","16128","1568","64"],["589824","2359296","3096576","2064384","806400","193536","28224","2304","81"],["2621440","11796480","17694720","13762560","6451200","1935360","376320","46080","3240","100"],["11534336","57671680","97320960","86507520","47308800","17031168","4139520","675840","71280","4400","121"],["50331648","276824064","519045120","519045120","324403200","136249344","39739392","8110080","1140480","105600","5808","144"],["218103808","1308622848","2699034624","2998927360","2108620800","1012137984","344408064","84344832","14826240","1830400","151008","7488","169"],["939524096","6106906624","13740539904","16793993216","13120307200","7084965888","2755264512","787218432","166053888","25625600","2818816","209664","9464","196"]],"degrees":[-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["0","1","12","105","800","5625","37500","240625","1500000","9140625","54687500","322265625","1875000000","10791015625","61523437500"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["1"]],"recurrence":{"denominator":{"coeffs":[["-4","1"],["4","-2"],["0","1"]]},"terms":[{"coeff":{"coeffs":[["0","0","0"],["16","4","0"],["0","4","1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":178794,"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-214d5046e1129b44"]},{"id":178795,"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-214d5046e1129b44"]},{"id":178796,"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-214d5046e1129b44"]}],"region_exact":{},"incoming_edges":[{"id":"e-214d5046e1129b44","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-033b1e898256e58f","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":"47079f29a8f32d61713328805cf5a730bca041cfbe126d6a7efe8727351f6ddd","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-ef5c9478e7d83dab","role":"sequence","name":"derivative of Triangle whose (i,j)-th entry is binomial(i,j)*4^(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-048305f2dd186ffc","kind":"fingerprint_related","node_a_id":"n-033b1e898256e58f","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-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-07cb3de7e9993649","other":{"id":"n-07cb3de7e9993649","name":"theta_plus_one of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^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":[["144","144","27"],["1728","2592","972","108"],["20736","41472","23328","5184","405"]],"card_rows":[["144","144","27"],["1728","2592","972","108"],["20736","41472","23328","5184","405"],["248832","622080","466560","155520","24300","1458"]],"row_sums_preview":["315","5400","91125","1518750","25059375","410062500","6663515625","107641406250"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-615df6ea54d3bbef","steps":["\u03b8+1"],"edges":["e-55c0a4c5884c9aec"],"text":"A038329 \u2192 \u03b8+1"},"headline":"A038329 \u2192 \u03b8+1","chain":null}},{"id":"l-5c556b58940b931f","kind":"fingerprint_related","node_a_id":"n-033b1e898256e58f","node_b_id":"n-f5b145ff37c5f0de","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-b702f561ac48283f","kind":"fingerprint_related","node_a_id":"n-033b1e898256e58f","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}},{"id":"l-cc41cfb6819a3c54","kind":"fingerprint_related","node_a_id":"n-033b1e898256e58f","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-dc48cd2fd0a1e6f0","kind":"fingerprint_related","node_a_id":"n-033b1e898256e58f","node_b_id":"n-7af05cad24952808","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-de03ca8c057368cd","kind":"fingerprint_related","node_a_id":"n-033b1e898256e58f","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-e7660609c529cc4b","kind":"fingerprint_related","node_a_id":"n-edc7bfe25fbf7be9","node_b_id":"n-033b1e898256e58f","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}}],"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:      8,       4;\n  3:     48,      48,       9;\n  4:    256,     384,     144,      16;\n  5:   1280,    2560,    1440,     320,     25;\n  6:   6144,   15360,   11520,    3840,    600,     36;\n  7:  28672,   86016,   80640,   35840,   8400,   1008,    49;\n  8: 131072,  458752,  516096,  286720,  89600,  16128,  1568,   64;\n  9: 589824, 2359296, 3096576, 2064384, 806400, 193536, 28224, 2304, 81;\n","recurrence_sentence":"The row polynomials satisfy (4*n-4+(n^2-2*n+1)*x)*P(n,x) = (16*n+(4*n^2+4*n)*x+n^2*x^2)*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[((16*n + (4*n^2 + 4*n)*x + n^2*x^2)*P[n - 1, x])/(4*n - 4 + (n^2 - 2*n + 1)*x)]];\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] = ((16*n + (4*n^2 + 4*n)*x + n^2*x^2)*v[i - 1])/(4*n - 4 + (n^2 - 2*n + 1)*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-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}