{"id":"n-87628f1b2cb96ac0","revision":0,"first_index":0,"fingerprint":"f6-57ca34e86b523c55","name":"derivative of basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*12^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":[["528","288"],["17424","19008","5184"],["511104","836352","456192","82944"],["14055360","30666240","25090560","9123840","1244160"],["371061504","1011985920","1103984640","602173440","164229120","17915904"],["9523911936","31169166336","42503408640","30911569920","12645642240","2759049216","250822656"],["239458357248","914295545856","1496119984128","1360109076480","741877678080","242796331008","44144787456","3439853568"],["5926594341888","25861502582784","49371959476224","53860319428608","36722945064960","16024557846528","4370333958144","681091006464","46438023168"],["144872306135040","711191321026560","1551690154967040","1974878379048960","1615809582858240","881350681559040","320491156930560","74920010711040","10216365096960","619173642240"],["3505909808467968","19123144409825280","46938627187752960","68274366818549760","65170986508615680","42657372987457536","19389714994298880","6043547530690560","1236180176732160","149840021422080","8173092077568"],["84141835403231232","504851012419387392","1376866397507420160","2253054105012142080","2457877205467791360","1876924411448131584","1023776951698980864","398874137025576960","108783855552430080","19778882827714560","2157696308477952","106993205379072"],["2005380410443677696","13126126322904072192","39378378968712216576","71597052670385848320","87869110095473541120","76685768810595090432","48800034697651421184","22815600637863002112","7778045671998750720","1885586829575454720","308550572112347136","30600056738414592","1390911669927936"],["47512089724357902336","336903908954537852928","1102594611123942064128","2205189222247884128256","3007076212156205629440","2952402099207910981632","2147201526696662532096","1171200832743634108416","479127613395123044352","145190185877310013440","31677858736867639296","4712408737715847168","428400794337804288","17974858503684096"],["1119927829217007697920","8552176150384422420480","30321351805908406763520","66155676667436523847680","99233515001154785771520","108254743637623402659840","88572062976237329448960","55213753543628465111040","26352018736731767439360","9582552267902460887040","2613423345791580241920","518364961148743188480","70686131065737707520","5931703306215751680","231105323618795520"],["26280973058959113977856","215026143209665478000640","821008910436904552366080","1940566515578138032865280","3175472480036953144688640","3810566976044343773626368","3464151796403948885114880","2429405155919652464885760","1325130085047083162664960","562176399716944372039680","183985003543727249031168","45616116581089400586240","8293839378379891015680","1043979781893972295680","81349073913816023040","2958148142320582656"]],"degrees":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],"row_sums":["816","41616","1886592","80180160","3271350528","129763570944","5042241613824","192865741728768","7286039131975680","272497863535890432","10107193483876663296","372281626656123764736","13631234945254993231872","496566415862860467732480","18008808681959739629764608"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["528","288"]],"recurrence":{"denominator":{"coeffs":[["1"],["1"]]},"terms":[{"coeff":{"coeffs":[["44","24"],["22","12"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":194608,"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-0413c38224a2b289"]},{"id":194609,"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-0413c38224a2b289"]},{"id":194610,"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-0413c38224a2b289"]},{"id":194611,"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-0413c38224a2b289"]}],"region_exact":{},"incoming_edges":[{"id":"e-0413c38224a2b289","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"derivative_polynomial\",\"coefficients\":[{\"numerator\":[\"0\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-87628f1b2cb96ac0","premise_count":1,"theorem":"Laguerre's theorem on differential operators: if phi is a real polynomial with only real roots, then phi(D) maps every real-rooted polynomial to a real-rooted polynomial or zero (write phi(D) = C D^j prod (1 + c_i D); D preserves real-rootedness by Rolle's theorem and p + c p' is real-rooted for every real c by Rolle's theorem applied to e^(x/c) p; Polya--Schur 1914); when the coefficients of phi have one sign, coefficients of one sign stay of one sign; applied to phi(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.splits_add_C_mul_derivative_all","RealRooted.splits_C_mul_add_derivative","Polynomial.Splits.derivative"],"certificate_hash":"d3df30e77c18ed4c4615ea78ff5874381386a21e202515a0863540fb901fe3e2","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-b536b63a5727c887","role":"sequence","name":"basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*12^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-0cff707239bb4db3","kind":"fingerprint_related","node_a_id":"n-1f2dec04776afb2b","node_b_id":"n-87628f1b2cb96ac0","fingerprint":"f6-57ca34e86b523c55","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-57ca34e86b523c55"},"other_id":"n-1f2dec04776afb2b","other":{"id":"n-1f2dec04776afb2b","name":"Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*6^j.","first_index":0,"fingerprint":"f6-57ca34e86b523c55","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["121","132","36"],["1331","2178","1188","216"],["14641","31944","26136","9504","1296"]],"card_rows":[["121","132","36"],["1331","2178","1188","216"],["14641","31944","26136","9504","1296"],["161051","439230","479160","261360","71280","7776"]],"row_sums_preview":["289","4913","83521","1419857","24137569","410338673","6975757441","118587876497"],"regions":["(-inf, 0)","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-1f2dec04776afb2b","steps":[],"edges":[],"text":"A038320"},"headline":"Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*6^j.","chain":null}},{"id":"l-3949c3f7bade0858","kind":"fingerprint_related","node_a_id":"n-12372a4647df96de","node_b_id":"n-87628f1b2cb96ac0","fingerprint":"f6-57ca34e86b523c55","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-57ca34e86b523c55"},"other_id":"n-12372a4647df96de","other":{"id":"n-12372a4647df96de","name":"derivative of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*6^j.","first_index":0,"fingerprint":"f6-57ca34e86b523c55","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["132","72"],["2178","2376","648"],["31944","52272","28512","5184"]],"card_rows":[["132","72"],["2178","2376","648"],["31944","52272","28512","5184"],["439230","958320","784080","285120","38880"]],"row_sums_preview":["204","5202","117912","2505630","51114852","1013777898","19696256304","376690901814"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-1f2dec04776afb2b","steps":["D"],"edges":["e-ec5ce3b79dc5f6c1"],"text":"A038320 \u2192 D"},"headline":"A038320 \u2192 D","chain":null}},{"id":"l-3fef31cbbe361ade","kind":"fingerprint_related","node_a_id":"n-eef41637184d467b","node_b_id":"n-87628f1b2cb96ac0","fingerprint":"f6-57ca34e86b523c55","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-57ca34e86b523c55"},"other_id":"n-eef41637184d467b","other":{"id":"n-eef41637184d467b","name":"reversal of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*6^j.","first_index":0,"fingerprint":"f6-57ca34e86b523c55","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["36","132","121"],["216","1188","2178","1331"],["1296","9504","26136","31944","14641"]],"card_rows":[["36","132","121"],["216","1188","2178","1331"],["1296","9504","26136","31944","14641"],["7776","71280","261360","479160","439230","161051"]],"row_sums_preview":["289","4913","83521","1419857","24137569","410338673","6975757441","118587876497"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-1f2dec04776afb2b","steps":["reverse"],"edges":["e-4c4703fcbf72f296"],"text":"A038320 \u2192 reverse"},"headline":"A038320 \u2192 reverse","chain":null}},{"id":"l-5cb2c0b260aeeae0","kind":"fingerprint_related","node_a_id":"n-87628f1b2cb96ac0","node_b_id":"n-bdf842f807fcf23f","fingerprint":"f6-57ca34e86b523c55","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-57ca34e86b523c55"},"other_id":"n-bdf842f807fcf23f","other":{"id":"n-bdf842f807fcf23f","name":"shift_x_plus_one of basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*8^j.","first_index":0,"fingerprint":"f6-57ca34e86b523c55","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1936","2112","576"],["85184","139392","76032","13824"],["3748096","8177664","6690816","2433024","331776"]],"card_rows":[["1936","2112","576"],["85184","139392","76032","13824"],["3748096","8177664","6690816","2433024","331776"],["164916224","449771520","490659840","267632640","72990720","7962624"]],"row_sums_preview":["4624","314432","21381376","1453933568","98867482624","6722988818432","457163239653376","31087100296429568"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-6874f381e9c7e0a2","steps":["basis A013610","basis A007318"],"edges":["e-889e30cc3b49f76a","e-57f28868c68f1b7f"],"text":"A038334 \u2192 basis A013610 \u2192 basis A007318"},"headline":"A038334 \u2192 basis A013610 \u2192 basis A007318","chain":null}},{"id":"l-7ca318ead27d9c51","kind":"fingerprint_related","node_a_id":"n-43a1e62341e55229","node_b_id":"n-87628f1b2cb96ac0","fingerprint":"f6-57ca34e86b523c55","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-57ca34e86b523c55"},"other_id":"n-43a1e62341e55229","other":{"id":"n-43a1e62341e55229","name":"basis_map_A013609 of Triangle of coefficients in expansion of (1+11x)^n.","first_index":0,"fingerprint":"f6-57ca34e86b523c55","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["144","528","484"],["1728","9504","17424","10648"],["20736","152064","418176","511104","234256"]],"card_rows":[["144","528","484"],["1728","9504","17424","10648"],["20736","152064","418176","511104","234256"],["248832","2280960","8363520","15333120","14055360","5153632"]],"row_sums_preview":["1156","39304","1336336","45435424","1544804416","52523350144","1785793904896","60716992766464"],"regions":["(-inf, 0]","[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ec89fb3fc20684fc","steps":["x \u21a6 x/(x+1)"],"edges":["e-492aa71fabef2dc6"],"text":"A038336 \u2192 x \u21a6 x/(x+1)"},"headline":"A038336 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-87683fb163af38db","kind":"fingerprint_related","node_a_id":"n-196d480a741a9e86","node_b_id":"n-87628f1b2cb96ac0","fingerprint":"f6-57ca34e86b523c55","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-57ca34e86b523c55"},"other_id":"n-196d480a741a9e86","other":{"id":"n-196d480a741a9e86","name":"basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*11^j.","first_index":0,"fingerprint":"f6-57ca34e86b523c55","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["18","33"],["324","1188","1089"]],"card_rows":[["1"],["18","33"],["324","1188","1089"],["5832","32076","58806","35937"]],"row_sums_preview":["1","51","2601","132651","6765201","345025251","17596287801","897410677851"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-b7cf68162d40a956","steps":["basis A013610"],"edges":["e-77c9a747c1207e17"],"text":"A038277 \u2192 basis A013610"},"headline":"A038277 \u2192 basis A013610","chain":null}},{"id":"l-8902f750bcd61d77","kind":"fingerprint_related","node_a_id":"n-c0a81f2faf0eede5","node_b_id":"n-87628f1b2cb96ac0","fingerprint":"f6-57ca34e86b523c55","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-57ca34e86b523c55"},"other_id":"n-c0a81f2faf0eede5","other":{"id":"n-c0a81f2faf0eede5","name":"mobius_x_over_x_plus_one of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*5^j.","first_index":0,"fingerprint":"f6-57ca34e86b523c55","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["12","22"],["144","528","484"]],"card_rows":[["1"],["12","22"],["144","528","484"],["1728","9504","17424","10648"]],"row_sums_preview":["1","34","1156","39304","1336336","45435424","1544804416","52523350144"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a6a633b989da1879","steps":["basis A013609","x \u21a6 x/(x+1)"],"edges":["e-e169cb33ba7b2c8c","e-abc8fe4c398b38d7"],"text":"A038271 \u2192 basis A013609 \u2192 x \u21a6 x/(x+1)"},"headline":"A038271 \u2192 basis A013609 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-aa36d2ce8010e62d","kind":"fingerprint_related","node_a_id":"n-739865e64d311853","node_b_id":"n-87628f1b2cb96ac0","fingerprint":"f6-57ca34e86b523c55","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-57ca34e86b523c55"},"other_id":"n-739865e64d311853","other":{"id":"n-739865e64d311853","name":"mobius_x_over_x_plus_one of basis_map_A007318 of Triangle of coefficients in expansion of (1+5x)^n.","first_index":0,"fingerprint":"f6-57ca34e86b523c55","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["36","132","121"],["216","1188","2178","1331"],["1296","9504","26136","31944","14641"]],"card_rows":[["36","132","121"],["216","1188","2178","1331"],["1296","9504","26136","31944","14641"],["7776","71280","261360","479160","439230","161051"]],"row_sums_preview":["289","4913","83521","1419857","24137569","410338673","6975757441","118587876497"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-bbfc5a54ff13a785","steps":["x \u21a6 x+1","x \u21a6 x/(x+1)"],"edges":["e-7ae18e4d322020fd","e-45bd992a18b3860c"],"text":"A013612 \u2192 x \u21a6 x+1 \u2192 x \u21a6 x/(x+1)"},"headline":"A013612 \u2192 x \u21a6 x+1 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-afa94ef71b2e095d","kind":"fingerprint_related","node_a_id":"n-8c6dd2889f1f26f5","node_b_id":"n-87628f1b2cb96ac0","fingerprint":"f6-57ca34e86b523c55","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-57ca34e86b523c55"},"other_id":"n-8c6dd2889f1f26f5","other":{"id":"n-8c6dd2889f1f26f5","name":"derivative of basis_map_A013609 of Triangle of coefficients in expansion of (1+11x)^n.","first_index":0,"fingerprint":"f6-57ca34e86b523c55","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["528","968"],["9504","34848","31944"],["152064","836352","1533312","937024"]],"card_rows":[["528","968"],["9504","34848","31944"],["152064","836352","1533312","937024"],["2280960","16727040","45999360","56221440","25768160"]],"row_sums_preview":["1496","76296","3458752","146996960","5997475968","237899880064","9244109625344","353587193169408"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ec89fb3fc20684fc","steps":["x \u21a6 x/(x+1)","D"],"edges":["e-492aa71fabef2dc6","e-f364e89c32463b67"],"text":"A038336 \u2192 x \u21a6 x/(x+1) \u2192 D"},"headline":"A038336 \u2192 x \u21a6 x/(x+1) \u2192 D","chain":null}},{"id":"l-ee63b893689c6921","kind":"fingerprint_related","node_a_id":"n-ebafadb809217503","node_b_id":"n-87628f1b2cb96ac0","fingerprint":"f6-57ca34e86b523c55","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-57ca34e86b523c55"},"other_id":"n-ebafadb809217503","other":{"id":"n-ebafadb809217503","name":"shift_x_plus_one of basis_map_A013609 of Triangle of coefficients in expansion of (2+3x)^n.","first_index":0,"fingerprint":"f6-57ca34e86b523c55","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["11","6"],["121","132","36"]],"card_rows":[["1"],["11","6"],["121","132","36"],["1331","2178","1188","216"]],"row_sums_preview":["1","17","289","4913","83521","1419857","24137569","410338673"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-f211cb6153c37c6d","steps":["basis A013610","basis A013609"],"edges":["e-892638ce5a781886","e-1c2af4d858ccc914"],"text":"A027466 \u2192 basis A013610 \u2192 basis A013609"},"headline":"A027466 \u2192 basis A013610 \u2192 basis A013609","chain":null}},{"id":"l-ef8fbf97c77dc01f","kind":"fingerprint_related","node_a_id":"n-1cf8aaa727e34c93","node_b_id":"n-87628f1b2cb96ac0","fingerprint":"f6-57ca34e86b523c55","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-57ca34e86b523c55"},"other_id":"n-1cf8aaa727e34c93","other":{"id":"n-1cf8aaa727e34c93","name":"mobius_x_over_x_plus_one of Triangle whose (i,j)-th entry is binomial(i,j)*6^(i-j)*5^j.","first_index":0,"fingerprint":"f6-57ca34e86b523c55","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["6","11"],["36","132","121"]],"card_rows":[["1"],["6","11"],["36","132","121"],["216","1188","2178","1331"]],"row_sums_preview":["1","17","289","4913","83521","1419857","24137569","410338673"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-e6ebe74518d42cbe","steps":["x \u21a6 x/(x+1)"],"edges":["e-afe3a450266622c8"],"text":"A038259 \u2192 x \u21a6 x/(x+1)"},"headline":"A038259 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-f979e013694b3d04","kind":"fingerprint_related","node_a_id":"n-b536b63a5727c887","node_b_id":"n-87628f1b2cb96ac0","fingerprint":"f6-57ca34e86b523c55","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-57ca34e86b523c55"},"other_id":"n-b536b63a5727c887","other":{"id":"n-b536b63a5727c887","name":"basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*12^j.","first_index":0,"fingerprint":"f6-57ca34e86b523c55","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["484","528","144"],["10648","17424","9504","1728"],["234256","511104","418176","152064","20736"]],"card_rows":[["484","528","144"],["10648","17424","9504","1728"],["234256","511104","418176","152064","20736"],["5153632","14055360","15333120","8363520","2280960","248832"]],"row_sums_preview":["1156","39304","1336336","45435424","1544804416","52523350144","1785793904896","60716992766464"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-29d7c5a1e19b9d74","steps":["x \u21a6 x+1"],"edges":["e-e77df5b989acd52a"],"text":"A038314 \u2192 x \u21a6 x+1"},"headline":"A038314 \u2192 x \u21a6 x+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:              528,               288;\n  1:            17424,             19008,              5184;\n  2:           511104,            836352,            456192,             82944;\n  3:         14055360,          30666240,          25090560,           9123840,           1244160;\n  4:        371061504,        1011985920,        1103984640,         602173440,         164229120,          17915904;\n  5:       9523911936,       31169166336,       42503408640,       30911569920,       12645642240,        2759049216,         250822656;\n  6:     239458357248,      914295545856,     1496119984128,     1360109076480,      741877678080,      242796331008,       44144787456,       3439853568;\n  7:    5926594341888,    25861502582784,    49371959476224,    53860319428608,    36722945064960,    16024557846528,     4370333958144,     681091006464,      46438023168;\n  8:  144872306135040,   711191321026560,  1551690154967040,  1974878379048960,  1615809582858240,   881350681559040,   320491156930560,   74920010711040,   10216365096960,    619173642240;\n  9: 3505909808467968, 19123144409825280, 46938627187752960, 68274366818549760, 65170986508615680, 42657372987457536, 19389714994298880, 6043547530690560, 1236180176732160, 149840021422080, 8173092077568;\n","recurrence_sentence":"The row polynomials satisfy (n+1)*P(n,x) = (22*n+44+(12*n+24)*x)*P(n-1,x) for n >= 1, with P(0,x) = 528 + 288*x.","mathematica":"P[0, x_] := 528 + 288*x;\nP[n_, x_] := P[n, x] = Expand[Together[((22*n + 44 + (12*n + 24)*x)*P[n - 1, x])/(n + 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 528 + 288*x; for(n = 1, N, my(i = n + 1); v[i] = ((22*n + 44 + (12*n + 24)*x)*v[i - 1])/(n + 1)); 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-29d7c5a1e19b9d74","steps":["x \u21a6 x+1","D"],"edges":["e-e77df5b989acd52a","e-0413c38224a2b289"],"text":"A038314 \u2192 x \u21a6 x+1 \u2192 D"},"headline":"A038314 \u2192 x \u21a6 x+1 \u2192 D","chain":null}