{"id":"n-76d5e94fd92ac146","revision":0,"first_index":0,"fingerprint":"f6-e620d4a356abb37c","name":"theta_plus_one of shift_x_plus_one of Exponential Riordan array [x*exp(x),x].","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","4"],["12","24","9"],["32","96","72","16"],["80","320","360","160","25"],["192","960","1440","960","300","36"],["448","2688","5040","4480","2100","504","49"],["1024","7168","16128","17920","11200","4032","784","64"],["2304","18432","48384","64512","50400","24192","7056","1152","81"],["5120","46080","138240","215040","201600","120960","47040","11520","1620","100"],["11264","112640","380160","675840","739200","532224","258720","84480","17820","2200","121"],["24576","270336","1013760","2027520","2534400","2128896","1241856","506880","142560","26400","2904","144"],["53248","638976","2635776","5857280","8236800","7907328","5381376","2635776","926640","228800","37752","3744","169"],["114688","1490944","6709248","16400384","25625600","27675648","21525504","12300288","5189184","1601600","352352","52416","4732","196"]],"degrees":[-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["0","1","8","45","216","945","3888","15309","58320","216513","787320","2814669","9920232","34543665","119042784"],"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":[["-2","1"],["2","-2"],["0","1"]]},"terms":[{"coeff":{"coeffs":[["0","0","0"],["4","2","0"],["0","2","1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":174583,"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-320b91005dd84266"]},{"id":174584,"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-320b91005dd84266"]},{"id":174585,"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-320b91005dd84266"]}],"region_exact":{},"incoming_edges":[{"id":"e-320b91005dd84266","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-76d5e94fd92ac146","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":"7c0a25588885d38af1bccbfd96121c0f510c30d68aa587523e65edb94b383976","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-7c4e320f16a7ccb1","role":"sequence","name":"shift_x_plus_one of Exponential Riordan array [x*exp(x),x].","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-00a04f5d4b0f674d","kind":"fingerprint_related","node_a_id":"n-76d5e94fd92ac146","node_b_id":"n-d03b3ca3ebd61dd6","fingerprint":"f6-e620d4a356abb37c","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e620d4a356abb37c"},"other_id":"n-d03b3ca3ebd61dd6","other":{"id":"n-d03b3ca3ebd61dd6","name":"theta_plus_one of basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*2^i.","first_index":0,"fingerprint":"f6-e620d4a356abb37c","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","4"],["16","32","12"]],"card_rows":[["1"],["4","4"],["16","32","12"],["64","192","144","32"]],"row_sums_preview":["1","8","60","432","3024","20736","139968","933120"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-d28ad9841fc73317","steps":["basis A007318","\u03b8+1"],"edges":["e-1756f066a00732e3","e-3802ca4dba8dabb6"],"text":"A038208 \u2192 basis A007318 \u2192 \u03b8+1"},"headline":"A038208 \u2192 basis A007318 \u2192 \u03b8+1","chain":null}},{"id":"l-04d4f42c38bafe8d","kind":"fingerprint_related","node_a_id":"n-76d5e94fd92ac146","node_b_id":"n-c5c9ce6bb3129f42","fingerprint":"f6-e620d4a356abb37c","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e620d4a356abb37c"},"other_id":"n-c5c9ce6bb3129f42","other":{"id":"n-c5c9ce6bb3129f42","name":"theta_plus_one of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j.","first_index":0,"fingerprint":"f6-e620d4a356abb37c","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["256","512","192"],["4096","12288","9216","2048"],["65536","262144","294912","131072","20480"]],"card_rows":[["256","512","192"],["4096","12288","9216","2048"],["65536","262144","294912","131072","20480"],["1048576","5242880","7864320","5242880","1638400","196608"]],"row_sums_preview":["960","27648","774144","21233664","573308928","15288238080","403609485312","10567230160896"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a93042cd70c4761e","steps":["basis A013609","\u03b8+1"],"edges":["e-6efc8fa5932928ce","e-a3fc1ee28267f768"],"text":"A038330 \u2192 basis A013609 \u2192 \u03b8+1"},"headline":"A038330 \u2192 basis A013609 \u2192 \u03b8+1","chain":null}},{"id":"l-14e27495b3581ac2","kind":"fingerprint_related","node_a_id":"n-76d5e94fd92ac146","node_b_id":"n-78f041e86f6d0ee1","fingerprint":"f6-e620d4a356abb37c","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e620d4a356abb37c"},"other_id":"n-78f041e86f6d0ee1","other":{"id":"n-78f041e86f6d0ee1","name":"basis_map_A013609 of one_plus_derivative of 3n + 1 preceded by n zeros.","first_index":0,"fingerprint":"f6-e620d4a356abb37c","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["8","8"],["21","56","28"]],"card_rows":[["1"],["8","8"],["21","56","28"],["40","180","240","80"]],"row_sums_preview":["1","16","105","540","2457","10368","41553","160380"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ccc86c3dcf2ac685","steps":["D+1","basis A013609"],"edges":["e-133ff869421f85a2","e-4dee996ff28df02f"],"text":"A131431 \u2192 D+1 \u2192 basis A013609"},"headline":"A131431 \u2192 D+1 \u2192 basis A013609","chain":null}},{"id":"l-1f2d9186b0c80452","kind":"fingerprint_related","node_a_id":"n-76d5e94fd92ac146","node_b_id":"n-d38ccd7c83577715","fingerprint":"f6-e620d4a356abb37c","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e620d4a356abb37c"},"other_id":"n-d38ccd7c83577715","other":{"id":"n-d38ccd7c83577715","name":"basis_map_A013609 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-e620d4a356abb37c","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["1"],["4","4"]],"card_rows":[["1"],["4","4"],["9","24","12"],["16","72","96","32"]],"row_sums_preview":["0","1","8","45","216","945","3888","15309"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-1c11ffc3c4cd47ed","steps":["D+1","basis A013609"],"edges":["e-9e777fb0c714fbf5","e-5711be6e2cb5d80a"],"text":"A132440 \u2192 D+1 \u2192 basis A013609"},"headline":"A132440 \u2192 D+1 \u2192 basis A013609","chain":null}},{"id":"l-2002042dce9dad47","kind":"fingerprint_related","node_a_id":"n-76d5e94fd92ac146","node_b_id":"n-4f58ee6499244b82","fingerprint":"f6-e620d4a356abb37c","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e620d4a356abb37c"},"other_id":"n-4f58ee6499244b82","other":{"id":"n-4f58ee6499244b82","name":"theta_plus_one of basis_map_A007318 of Triangle T(n, k) read by rows, T(n, k) = n!*binomial(n, k).","first_index":0,"fingerprint":"f6-e620d4a356abb37c","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","2"],["8","16","6"]],"card_rows":[["1"],["2","2"],["8","16","6"],["48","144","108","24"]],"row_sums_preview":["1","4","30","324","4536","77760","1574640","36741600"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-cb5d49e71e46f2ed","steps":["x \u21a6 x+1","\u03b8+1"],"edges":["e-0b89a3d3e0cfd72c","e-3776116722ba1085"],"text":"A196347 \u2192 x \u21a6 x+1 \u2192 \u03b8+1"},"headline":"A196347 \u2192 x \u21a6 x+1 \u2192 \u03b8+1","chain":null}},{"id":"l-37defb013a16de35","kind":"fingerprint_related","node_a_id":"n-09791955a4a02550","node_b_id":"n-76d5e94fd92ac146","fingerprint":"f6-e620d4a356abb37c","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e620d4a356abb37c"},"other_id":"n-09791955a4a02550","other":{"id":"n-09791955a4a02550","name":"theta_plus_one of Triangle whose (i,j)-th entry is binomial(i,j)*2^(i-j).","first_index":0,"fingerprint":"f6-e620d4a356abb37c","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","2"],["4","8","3"]],"card_rows":[["1"],["2","2"],["4","8","3"],["8","24","18","4"]],"row_sums_preview":["1","4","15","54","189","648","2187","7290"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-af754f07daf65b5e","steps":["\u03b8+1"],"edges":["e-78274d1fdc91ece9"],"text":"A038207 \u2192 \u03b8+1"},"headline":"A038207 \u2192 \u03b8+1","chain":null}},{"id":"l-38fd7086dbb0bec9","kind":"fingerprint_related","node_a_id":"n-76d5e94fd92ac146","node_b_id":"n-042d5cf9e98e7765","fingerprint":"f6-e620d4a356abb37c","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e620d4a356abb37c"},"other_id":"n-042d5cf9e98e7765","other":{"id":"n-042d5cf9e98e7765","name":"theta_plus_one of basis_map_A007318 of Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by the n-th Catalan number (A000108).","first_index":1,"fingerprint":"f6-e620d4a356abb37c","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","2"],["8","16","6"]],"card_rows":[["1"],["2","2"],["8","16","6"],["40","120","90","20"]],"row_sums_preview":["1","4","30","270","2646","27216","288684","3127410"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a98fae1bf9eeb913","steps":["x \u21a6 x+1","\u03b8+1"],"edges":["e-45645d17dad7cf5c","e-eefced593184c8ed"],"text":"A085880 \u2192 x \u21a6 x+1 \u2192 \u03b8+1"},"headline":"A085880 \u2192 x \u21a6 x+1 \u2192 \u03b8+1","chain":null}},{"id":"l-3dd297d7d7388a0c","kind":"fingerprint_related","node_a_id":"n-76d5e94fd92ac146","node_b_id":"n-3b2bfb7c3621f647","fingerprint":"f6-e620d4a356abb37c","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e620d4a356abb37c"},"other_id":"n-3b2bfb7c3621f647","other":{"id":"n-3b2bfb7c3621f647","name":"theta_plus_one of basis_map_A007318 of Triangle read by rows: T(n,k) = binomial(n+4,4) * binomial(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-e620d4a356abb37c","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["10","10"],["60","120","45"]],"card_rows":[["1"],["10","10"],["60","120","45"],["280","840","630","140"]],"row_sums_preview":["1","20","225","1890","13230","81648","459270","2405700"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-79986ab5023c2764","steps":["basis A007318","\u03b8+1"],"edges":["e-484d449f4d0f99c5","e-a56bd583e162a2f4"],"text":"A178821 \u2192 basis A007318 \u2192 \u03b8+1"},"headline":"A178821 \u2192 basis A007318 \u2192 \u03b8+1","chain":null}},{"id":"l-41313effa2d1e797","kind":"fingerprint_related","node_a_id":"n-76d5e94fd92ac146","node_b_id":"n-ee264dcf67d468d3","fingerprint":"f6-e620d4a356abb37c","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e620d4a356abb37c"},"other_id":"n-ee264dcf67d468d3","other":{"id":"n-ee264dcf67d468d3","name":"theta_plus_one of basis_map_A007318 of Triangle read by rows: T(n,k) = C(n+5,5) * C(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-e620d4a356abb37c","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["12","12"],["84","168","63"]],"card_rows":[["1"],["12","12"],["84","168","63"],["448","1344","1008","224"]],"row_sums_preview":["1","24","315","3024","23814","163296","1010394","5773680"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-97228a88c81561dc","steps":["x \u21a6 x+1","\u03b8+1"],"edges":["e-a52727d7b51dd683","e-16f9881d10bb270c"],"text":"A178822 \u2192 x \u21a6 x+1 \u2192 \u03b8+1"},"headline":"A178822 \u2192 x \u21a6 x+1 \u2192 \u03b8+1","chain":null}},{"id":"l-4a621fd292271677","kind":"fingerprint_related","node_a_id":"n-4ec97bc875f9c45e","node_b_id":"n-76d5e94fd92ac146","fingerprint":"f6-e620d4a356abb37c","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e620d4a356abb37c"},"other_id":"n-4ec97bc875f9c45e","other":{"id":"n-4ec97bc875f9c45e","name":"theta_plus_one 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-e620d4a356abb37c","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["3"],["6","6"],["12","24","9"]],"card_rows":[["3"],["6","6"],["12","24","9"],["24","72","54","12"]],"row_sums_preview":["3","12","45","162","567","1944","6561","21870"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-51b4f5e7901c2b50","steps":["\u03b8+1"],"edges":["e-217a3f649e225959"],"text":"A120910 \u2192 \u03b8+1"},"headline":"A120910 \u2192 \u03b8+1","chain":null}},{"id":"l-4ffcab4666915359","kind":"fingerprint_related","node_a_id":"n-76d5e94fd92ac146","node_b_id":"n-141ca264cc04332e","fingerprint":"f6-e620d4a356abb37c","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e620d4a356abb37c"},"other_id":"n-141ca264cc04332e","other":{"id":"n-141ca264cc04332e","name":"theta_plus_one of basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*3^j.","first_index":0,"fingerprint":"f6-e620d4a356abb37c","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["6","6"],["36","72","27"]],"card_rows":[["1"],["6","6"],["36","72","27"],["216","648","486","108"]],"row_sums_preview":["1","12","135","1458","15309","157464","1594323","15943230"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-8d88ef9de74b305f","steps":["x \u21a6 x+1","\u03b8+1"],"edges":["e-02e8db4659f506e0","e-7f636cfd66255339"],"text":"A038221 \u2192 x \u21a6 x+1 \u2192 \u03b8+1"},"headline":"A038221 \u2192 x \u21a6 x+1 \u2192 \u03b8+1","chain":null}},{"id":"l-59c1bcd96bcaf098","kind":"fingerprint_related","node_a_id":"n-76d5e94fd92ac146","node_b_id":"n-3cad8bd53801215d","fingerprint":"f6-e620d4a356abb37c","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e620d4a356abb37c"},"other_id":"n-3cad8bd53801215d","other":{"id":"n-3cad8bd53801215d","name":"theta_plus_one of basis_map_A007318 of Triangle read by rows: T(n,k) = C(n+3,3) * C(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-e620d4a356abb37c","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["8","8"],["40","80","30"]],"card_rows":[["1"],["8","8"],["40","80","30"],["160","480","360","80"]],"row_sums_preview":["1","16","150","1080","6615","36288","183708","874800"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a820c1859f68e71c","steps":["basis A007318","\u03b8+1"],"edges":["e-2fa6de2302a1c958","e-daf45bc96b7e0aeb"],"text":"A178820 \u2192 basis A007318 \u2192 \u03b8+1"},"headline":"A178820 \u2192 basis A007318 \u2192 \u03b8+1","chain":null}},{"id":"l-634d8d29ffe573af","kind":"fingerprint_related","node_a_id":"n-76d5e94fd92ac146","node_b_id":"n-fbe3a8180125f580","fingerprint":"f6-e620d4a356abb37c","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e620d4a356abb37c"},"other_id":"n-fbe3a8180125f580","other":{"id":"n-fbe3a8180125f580","name":"theta_plus_one of shift_x_plus_one of Triangle read by rows: T(n,k) = ((n+1)(n+2)/2) * binomial(n,k) (0 <= k <= n).","first_index":0,"fingerprint":"f6-e620d4a356abb37c","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["6","6"],["24","48","18"]],"card_rows":[["1"],["6","6"],["24","48","18"],["80","240","180","40"]],"row_sums_preview":["1","12","90","540","2835","13608","61236","262440"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7f1b09730105c8f0","steps":["x \u21a6 x+1","\u03b8+1"],"edges":["e-34e3d9a08d46fb54","e-5a40445eda7fefdf"],"text":"A094305 \u2192 x \u21a6 x+1 \u2192 \u03b8+1"},"headline":"A094305 \u2192 x \u21a6 x+1 \u2192 \u03b8+1","chain":null}},{"id":"l-71f4e7f5ac483f08","kind":"fingerprint_related","node_a_id":"n-76d5e94fd92ac146","node_b_id":"n-439dbbff0fdd48fb","fingerprint":"f6-e620d4a356abb37c","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e620d4a356abb37c"},"other_id":"n-439dbbff0fdd48fb","other":{"id":"n-439dbbff0fdd48fb","name":"basis_map_A013609 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-e620d4a356abb37c","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["8","8"],["27","72","36"]],"card_rows":[["1"],["8","8"],["27","72","36"],["64","288","384","128"]],"row_sums_preview":["1","16","135","864","4725","23328","107163","466560"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7b3743413970d91c","steps":["\u03b8+1","D+1","basis A013609"],"edges":["e-06738ea5bb413735","e-2dc8b6d950a061ae","e-446a0b469defadbd"],"text":"image under \u03b8+1 \u2192 \u03b8+1 \u2192 D+1 \u2192 basis A013609"},"headline":"image under \u03b8+1 \u2192 \u03b8+1 \u2192 D+1 \u2192 basis A013609","chain":null}},{"id":"l-79e48a30f5ed04f9","kind":"fingerprint_related","node_a_id":"n-76d5e94fd92ac146","node_b_id":"n-b1b7a75e45782fe8","fingerprint":"f6-e620d4a356abb37c","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e620d4a356abb37c"},"other_id":"n-b1b7a75e45782fe8","other":{"id":"n-b1b7a75e45782fe8","name":"theta_plus_one of basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j).","first_index":0,"fingerprint":"f6-e620d4a356abb37c","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["8","8"],["64","128","48"]],"card_rows":[["1"],["8","8"],["64","128","48"],["512","1536","1152","256"]],"row_sums_preview":["1","16","240","3456","48384","663552","8957952","119439360"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-f211cb6153c37c6d","steps":["basis A013611","\u03b8+1"],"edges":["e-267942f18014ef6e","e-280008cbbdc3bb87"],"text":"A027466 \u2192 basis A013611 \u2192 \u03b8+1"},"headline":"A027466 \u2192 basis A013611 \u2192 \u03b8+1","chain":null}},{"id":"l-864a2eb9438d8da5","kind":"fingerprint_related","node_a_id":"n-76d5e94fd92ac146","node_b_id":"n-78f7c675f6d6e8e6","fingerprint":"f6-e620d4a356abb37c","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e620d4a356abb37c"},"other_id":"n-78f7c675f6d6e8e6","other":{"id":"n-78f7c675f6d6e8e6","name":"theta_plus_one of basis_map_A007318 of Triangle read by rows whose (i,j)-th entry is binomial(i,j)*11^(i-j)*11^j.","first_index":0,"fingerprint":"f6-e620d4a356abb37c","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["22","22"],["484","968","363"]],"card_rows":[["1"],["22","22"],["484","968","363"],["10648","31944","23958","5324"]],"row_sums_preview":["1","44","1815","71874","2767149","104361048","3874403907","142061476590"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-8064e72cf80883a7","steps":["basis A007318","\u03b8+1"],"edges":["e-a5d0659f5315d3b7","e-9dfb5436cc1607cd"],"text":"A038325 \u2192 basis A007318 \u2192 \u03b8+1"},"headline":"A038325 \u2192 basis A007318 \u2192 \u03b8+1","chain":null}},{"id":"l-9dbc6a646911481a","kind":"fingerprint_related","node_a_id":"n-ea204d38c8ad3e29","node_b_id":"n-76d5e94fd92ac146","fingerprint":"f6-e620d4a356abb37c","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e620d4a356abb37c"},"other_id":"n-ea204d38c8ad3e29","other":{"id":"n-ea204d38c8ad3e29","name":"theta_plus_one of shift_x_plus_one of Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).","first_index":0,"fingerprint":"f6-e620d4a356abb37c","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["2"],["4","4"],["8","16","6"]],"card_rows":[["2"],["4","4"],["8","16","6"],["16","48","36","8"]],"row_sums_preview":["2","8","30","108","378","1296","4374","14580"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-089a9b4f824a1c20","steps":["basis A007318","\u03b8+1"],"edges":["e-014dd8a0f9594ad5","e-5d6c1edf94395a1f"],"text":"A028326 \u2192 basis A007318 \u2192 \u03b8+1"},"headline":"A028326 \u2192 basis A007318 \u2192 \u03b8+1","chain":null}},{"id":"l-a458736cad319e45","kind":"fingerprint_related","node_a_id":"n-ac13da29bb536a2b","node_b_id":"n-76d5e94fd92ac146","fingerprint":"f6-e620d4a356abb37c","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e620d4a356abb37c"},"other_id":"n-ac13da29bb536a2b","other":{"id":"n-ac13da29bb536a2b","name":"theta_plus_one of shift_x_plus_one of basis-map coefficients of Riordan array (1+2x,x(1+x)). shifted by x -> -x + 0","first_index":0,"fingerprint":"f6-e620d4a356abb37c","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":1,"preview_rows":[["1"],["-2","-2"],["4","8","3"]],"card_rows":[["1"],["-2","-2"],["4","8","3"],["-8","-24","-18","-4"]],"row_sums_preview":["1","-4","15","-54","189","-648","2187","-7290"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-74c1e94669ff21ff","steps":["x \u21a6 x+1","\u03b8+1"],"edges":["e-51a320c1f4e00092","e-290d87cad4bf8469"],"text":"basis-map coefficients of A113214, shifted \u2192 x \u21a6 x+1 \u2192 \u03b8+1"},"headline":"basis-map coefficients of A113214, shifted \u2192 x \u21a6 x+1 \u2192 \u03b8+1","chain":null}},{"id":"l-b8b188755fb642c3","kind":"fingerprint_related","node_a_id":"n-76d5e94fd92ac146","node_b_id":"n-706c0d1ff0d7787a","fingerprint":"f6-e620d4a356abb37c","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e620d4a356abb37c"},"other_id":"n-706c0d1ff0d7787a","other":{"id":"n-706c0d1ff0d7787a","name":"theta_plus_one of basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*12^j.","first_index":0,"fingerprint":"f6-e620d4a356abb37c","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["576","1152","432"],["13824","41472","31104","6912"],["331776","1327104","1492992","663552","103680"]],"card_rows":[["576","1152","432"],["13824","41472","31104","6912"],["331776","1327104","1492992","663552","103680"],["7962624","39813120","59719680","39813120","12441600","1492992"]],"row_sums_preview":["2160","93312","3919104","161243136","6530347008","261213880320","10344069660672","406239826673664"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c1532953a6736702","steps":["x \u21a6 x+1","\u03b8+1"],"edges":["e-240272ddc3b4d83d","e-cd109515d50286ca"],"text":"A038338 \u2192 x \u21a6 x+1 \u2192 \u03b8+1"},"headline":"A038338 \u2192 x \u21a6 x+1 \u2192 \u03b8+1","chain":null}},{"id":"l-ba0c64d411b68ff7","kind":"fingerprint_related","node_a_id":"n-76d5e94fd92ac146","node_b_id":"n-8ac1f80539d171bd","fingerprint":"f6-e620d4a356abb37c","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e620d4a356abb37c"},"other_id":"n-8ac1f80539d171bd","other":{"id":"n-8ac1f80539d171bd","name":"theta_plus_one of basis_map_A007318 of Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by A001147(n).","first_index":0,"fingerprint":"f6-e620d4a356abb37c","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","2"],["12","24","9"]],"card_rows":[["1"],["2","2"],["12","24","9"],["120","360","270","60"]],"row_sums_preview":["1","4","45","810","19845","612360","22733865","985134150"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c7809537965f400d","steps":["basis A007318","\u03b8+1"],"edges":["e-5048ce3ebd740a6f","e-5376bb6040cc5619"],"text":"A085881 \u2192 basis A007318 \u2192 \u03b8+1"},"headline":"A085881 \u2192 basis A007318 \u2192 \u03b8+1","chain":null}},{"id":"l-d0aa38cefee6f938","kind":"fingerprint_related","node_a_id":"n-76d5e94fd92ac146","node_b_id":"n-583419ea7c370e8a","fingerprint":"f6-e620d4a356abb37c","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e620d4a356abb37c"},"other_id":"n-583419ea7c370e8a","other":{"id":"n-583419ea7c370e8a","name":"theta_plus_one of basis_map_A007318 of Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.","first_index":0,"fingerprint":"f6-e620d4a356abb37c","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","4"],["24","48","18"]],"card_rows":[["1"],["4","4"],["24","48","18"],["192","576","432","96"]],"row_sums_preview":["1","8","90","1296","22680","466560","11022480","293932800"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-75a6475b3e851dc5","steps":["basis A007318","\u03b8+1"],"edges":["e-bd56c468ffc76132","e-a57ae7470d4aef5d"],"text":"A156992 \u2192 basis A007318 \u2192 \u03b8+1"},"headline":"A156992 \u2192 basis A007318 \u2192 \u03b8+1","chain":null}},{"id":"l-d57a4e4d4294a76c","kind":"fingerprint_related","node_a_id":"n-76d5e94fd92ac146","node_b_id":"n-f61f0ca91e0b2471","fingerprint":"f6-e620d4a356abb37c","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e620d4a356abb37c"},"other_id":"n-f61f0ca91e0b2471","other":{"id":"n-f61f0ca91e0b2471","name":"theta_plus_one of basis_map_A007318 of Triangle read by rows. T(n, k) = RisingFactorial(n + 1, n) / (k! * (n - k)!).","first_index":0,"fingerprint":"f6-e620d4a356abb37c","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","4"],["24","48","18"]],"card_rows":[["1"],["4","4"],["24","48","18"],["160","480","360","80"]],"row_sums_preview":["1","8","90","1080","13230","163296","2020788","25019280"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-dd77bf712b1df996","steps":["x \u21a6 x+1","\u03b8+1"],"edges":["e-0f8327ed3206c9f7","e-9b79204994697432"],"text":"A356546 \u2192 x \u21a6 x+1 \u2192 \u03b8+1"},"headline":"A356546 \u2192 x \u21a6 x+1 \u2192 \u03b8+1","chain":null}},{"id":"l-e5957c9c20a81e6f","kind":"fingerprint_related","node_a_id":"n-76d5e94fd92ac146","node_b_id":"n-ca86ef02eb5dd284","fingerprint":"f6-e620d4a356abb37c","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e620d4a356abb37c"},"other_id":"n-ca86ef02eb5dd284","other":{"id":"n-ca86ef02eb5dd284","name":"theta_plus_one of derivative 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-e620d4a356abb37c","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["3"],["12","12"]],"card_rows":[["3"],["12","12"],["36","72","27"],["96","288","216","48"]],"row_sums_preview":["0","3","24","135","648","2835","11664","45927"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-51b4f5e7901c2b50","steps":["D","\u03b8+1"],"edges":["e-d19cf3982d12ec2e","e-a4b91f9a8c1ed073"],"text":"A120910 \u2192 D \u2192 \u03b8+1"},"headline":"A120910 \u2192 D \u2192 \u03b8+1","chain":null}},{"id":"l-e73945cb2f4d8782","kind":"fingerprint_related","node_a_id":"n-76d5e94fd92ac146","node_b_id":"n-ba747c7b819ac5dd","fingerprint":"f6-e620d4a356abb37c","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e620d4a356abb37c"},"other_id":"n-ba747c7b819ac5dd","other":{"id":"n-ba747c7b819ac5dd","name":"theta_plus_one of shift_x_plus_one of A127733 * A007318 as infinite lower triangular matrices.","first_index":0,"fingerprint":"f6-e620d4a356abb37c","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["8","8"],["36","72","27"]],"card_rows":[["1"],["8","8"],["36","72","27"],["128","384","288","64"]],"row_sums_preview":["1","16","135","864","4725","23328","107163","466560"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-375ba152c4aebf61","steps":["x \u21a6 x+1","\u03b8+1"],"edges":["e-18ace392e620d7bd","e-375aebe05a8b241b"],"text":"A135065 \u2192 x \u21a6 x+1 \u2192 \u03b8+1"},"headline":"A135065 \u2192 x \u21a6 x+1 \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,     4;\n  3:   12,    24,     9;\n  4:   32,    96,    72,    16;\n  5:   80,   320,   360,   160,    25;\n  6:  192,   960,  1440,   960,   300,    36;\n  7:  448,  2688,  5040,  4480,  2100,   504,   49;\n  8: 1024,  7168, 16128, 17920, 11200,  4032,  784,   64;\n  9: 2304, 18432, 48384, 64512, 50400, 24192, 7056, 1152, 81;\n","recurrence_sentence":"The row polynomials satisfy (2*n-2+(n^2-2*n+1)*x)*P(n,x) = (4*n+(2*n^2+2*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[((4*n + (2*n^2 + 2*n)*x + n^2*x^2)*P[n - 1, x])/(2*n - 2 + (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] = ((4*n + (2*n^2 + 2*n)*x + n^2*x^2)*v[i - 1])/(2*n - 2 + (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-7b5887f058255aa1","steps":["x \u21a6 x+1","\u03b8+1"],"edges":["e-1a916fd0c60fb95f","e-320b91005dd84266"],"text":"A003506 \u2192 x \u21a6 x+1 \u2192 \u03b8+1"},"headline":"A003506 \u2192 x \u21a6 x+1 \u2192 \u03b8+1","chain":null}
