{"id":"n-8ea4f8a23684cfb4","revision":0,"first_index":0,"fingerprint":"f6-af50e1602aec03e4","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.","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"],["4","8"],["9","48","48"],["16","144","384","256"],["25","320","1440","2560","1280"],["36","600","3840","11520","15360","6144"],["49","1008","8400","35840","80640","86016","28672"],["64","1568","16128","89600","286720","516096","458752","131072"],["81","2304","28224","193536","806400","2064384","3096576","2359296","589824"],["100","3240","46080","376320","1935360","6451200","13762560","17694720","11796480","2621440"],["121","4400","71280","675840","4139520","17031168","47308800","86507520","97320960","57671680","11534336"],["144","5808","105600","1140480","8110080","39739392","136249344","324403200","519045120","519045120","276824064","50331648"],["169","7488","151008","1830400","14826240","84344832","344408064","1012137984","2108620800","2998927360","2699034624","1308622848","218103808"],["196","9464","209664","2818816","25625600","166053888","787218432","2755264512","7084965888","13120307200","16793993216","13740539904","6106906624","939524096"]],"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:basis_map_A013611","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["1"],["4","8"]],"recurrence":{"denominator":{"coeffs":[["2"],["-3"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","0","0"],["1","8","16"],["-1","-8","-16"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0"],["-4","-16"],["2","8"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":204215,"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-53401cb7af1f1125"]},{"id":204216,"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-53401cb7af1f1125"]},{"id":204217,"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-53401cb7af1f1125"]}],"region_exact":{},"incoming_edges":[{"id":"e-53401cb7af1f1125","operation":{"map":"A013611","operation":"basis_map"},"operation_detail":"A013611","conclusion_id":"n-8ea4f8a23684cfb4","premise_count":1,"theorem":"geometric family P_k = g(t) (a + d t)^k: T(p) = g(t) p(a + d t), so the roots of T(p) are those of g and the points (r - a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"8a9fe9aeb2327834487d245bce932ad569a3f5151eabffbb9408f7f4cee35fb3","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-d16445f4fa1fddba","role":"sequence","name":"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.","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-033b1e898256e58f","other":{"id":"n-033b1e898256e58f","name":"theta_plus_one of derivative 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"],["8","4"]],"card_rows":[["1"],["8","4"],["48","48","9"],["256","384","144","16"]],"row_sums_preview":["0","1","12","105","800","5625","37500","240625"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"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}},{"id":"l-3abf92cce2f68815","kind":"fingerprint_related","node_a_id":"n-8ea4f8a23684cfb4","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-4326ddee9a62ecba","kind":"fingerprint_related","node_a_id":"n-8ea4f8a23684cfb4","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-4f17676146b2dbd0","kind":"fingerprint_related","node_a_id":"n-f5b145ff37c5f0de","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-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-538af58f89fec1ee","kind":"fingerprint_related","node_a_id":"n-defea092f45b33f9","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-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-988ebcff942b78f8","kind":"fingerprint_related","node_a_id":"n-edc7bfe25fbf7be9","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-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}},{"id":"l-9fb0355e83bbb20d","kind":"fingerprint_related","node_a_id":"n-07cb3de7e9993649","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-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-e93abd10c25192a9","kind":"fingerprint_related","node_a_id":"n-7af05cad24952808","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-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}}],"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:  4,    8;\n  3:  9,   48,    48;\n  4: 16,  144,   384,    256;\n  5: 25,  320,  1440,   2560,   1280;\n  6: 36,  600,  3840,  11520,  15360,    6144;\n  7: 49, 1008,  8400,  35840,  80640,   86016,   28672;\n  8: 64, 1568, 16128,  89600, 286720,  516096,  458752,  131072;\n  9: 81, 2304, 28224, 193536, 806400, 2064384, 3096576, 2359296, 589824;\n","recurrence_sentence":"The row polynomials satisfy (n^2-3*n+2)*P(n,x) = -(n^2-n+(8*n^2-8*n)*x+(16*n^2-16*n)*x^2)*P(n-2,x) + (2*n^2-4*n+(8*n^2-16*n)*x)*P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 1, P(2,x) = 4 + 8*x.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 1;\nP[2, x_] := 4 + 8*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^2 - n + (8*n^2 - 8*n)*x + (16*n^2 - 16*n)*x^2)*P[n - 2, x] + (2*n^2 - 4*n + (8*n^2 - 16*n)*x)*P[n - 1, x])/(n^2 - 3*n + 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; v[3] = 4 + 8*x; for(n = 3, N, my(i = n + 1); v[i] = (-(n^2 - n + (8*n^2 - 8*n)*x + (16*n^2 - 16*n)*x^2)*v[i - 2] + (2*n^2 - 4*n + (8*n^2 - 16*n)*x)*v[i - 1])/(n^2 - 3*n + 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-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}