{"id":"n-fbee24f9afac6b9c","revision":0,"first_index":0,"fingerprint":"f6-42b2d489dcb16cdd","name":"basis_map_A013609 of basis_map_A007318 of Triangle of coefficients in expansion of (1+6x)^n.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[["1"],["13","12"],["169","312","144"],["2197","6084","5616","1728"],["28561","105456","146016","89856","20736"],["371293","1713660","3163680","2920320","1347840","248832"],["4826809","26733096","61691760","75928320","52565760","19408896","2985984"],["62748517","405451956","1122790032","1727369280","1594494720","883104768","271724544","35831808"],["815730721","6023857632","19461693888","35929281024","41456862720","30614298624","14129676288","3726508032","429981696"],["10604499373","88098917868","325288312128","700620979968","970090587648","895468234752","551057375232","218000719872","50307858432","5159780352"],["137858491849","1272539924760","5285935072080","13011532485120","21018629399040","23282174103552","17909364695040","9446697861120","3270010798080","670771445760","61917364224"],["1792160394037","18197320924068","83987635034160","232581143171520","429380572008960","554891816134656","512207830278144","337719448535040","155870514708480","47960158371840","8854183084032","743008370688"],["23298085122481","258071096741328","1310207106532896","4031406481639680","8372921154174720","12366160473858048","13317403587231744","10536846794293248","6078950073630720","2493928235335680","690626280554496","115909305827328","8916100448256"],["302875106592253","3634501279107036","20129545545823584","68130769539710592","157224852783947520","261235140010251264","321520172320309248","296787851372593152","205468512488718336","105368467942932480","38905280471236608","9794336342409216","1506820975755264","106993205379072"],["3937376385699289","50883017907498504","305298107444991024","1127254550566120704","2861492320667844864","5282755053540636672","7314583920287035392","7716484135687421952","6232544878824456192","3835412233122742272","1770190261441265664","594189738106159104","137120708793729024","19472763378991104","1283918464548864"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","25","625","15625","390625","9765625","244140625","6103515625","152587890625","3814697265625","95367431640625","2384185791015625","59604644775390625","1490116119384765625","37252902984619140625"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013609","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["13","12"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":200571,"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-1e50496cc65ef5b9","e-8510b4315777a543","e-c660dd3d30ecea2c"]},{"id":200572,"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-1e50496cc65ef5b9","e-8510b4315777a543","e-c660dd3d30ecea2c"]},{"id":200573,"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-1e50496cc65ef5b9","e-8510b4315777a543","e-c660dd3d30ecea2c"]}],"region_exact":{},"incoming_edges":[{"id":"e-1e50496cc65ef5b9","operation":{"map":"A013610","operation":"basis_map"},"operation_detail":"A013610","conclusion_id":"n-fbee24f9afac6b9c","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":"0c9fa415ee3629c23dbff6f4f22f2c19a313428dedd1e1336b1b37ad605d4f03","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-b8c0cc94a00a4dd8","role":"sequence","name":"basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*1^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]"}]},{"id":"e-8510b4315777a543","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-fbee24f9afac6b9c","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":"72a5e750649e14ac14f21ae0e8e9a2b16997d6b66e7d1556102388ea14c24c44","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-bc97c3236a99a08c","role":"sequence","name":"basis_map_A007318 of Triangle of coefficients in expansion of (1+6x)^n.","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]"}]},{"id":"e-c660dd3d30ecea2c","operation":{"map":"A013611","operation":"basis_map"},"operation_detail":"A013611","conclusion_id":"n-fbee24f9afac6b9c","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":"fcce4b21534d2149f3727845226f8b6e8c1870e549e148e324327b3c528433b1","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-6ca7f61b6ad83377","role":"sequence","name":"basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*9^(i-j)*1^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-1de7bace105a6213","kind":"fingerprint_related","node_a_id":"n-fbee24f9afac6b9c","node_b_id":"n-232fd19b6a19d182","fingerprint":"f6-42b2d489dcb16cdd","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-42b2d489dcb16cdd"},"other_id":"n-232fd19b6a19d182","other":{"id":"n-232fd19b6a19d182","name":"basis_map_A013609 of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*9^j.","first_index":0,"fingerprint":"f6-42b2d489dcb16cdd","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["39","36"],["1521","2808","1296"]],"card_rows":[["1"],["39","36"],["1521","2808","1296"],["59319","164268","151632","46656"]],"row_sums_preview":["1","75","5625","421875","31640625","2373046875","177978515625","13348388671875"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-f485cd65a8f8089f","steps":["basis A013609","basis A013609"],"edges":["e-f8a7b8e38f7708ee","e-956c9dfede11e18e"],"text":"A038335 \u2192 basis A013609 \u2192 basis A013609"},"headline":"A038335 \u2192 basis A013609 \u2192 basis A013609","chain":null}},{"id":"l-7db1216c8256804b","kind":"fingerprint_related","node_a_id":"n-b8a3f88eab200094","node_b_id":"n-fbee24f9afac6b9c","fingerprint":"f6-42b2d489dcb16cdd","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-42b2d489dcb16cdd"},"other_id":"n-b8a3f88eab200094","other":{"id":"n-b8a3f88eab200094","name":"basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*3^j.","first_index":0,"fingerprint":"f6-42b2d489dcb16cdd","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["169","312","144"],["2197","6084","5616","1728"],["28561","105456","146016","89856","20736"]],"card_rows":[["169","312","144"],["2197","6084","5616","1728"],["28561","105456","146016","89856","20736"],["371293","1713660","3163680","2920320","1347840","248832"]],"row_sums_preview":["625","15625","390625","9765625","244140625","6103515625","152587890625","3814697265625"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-4397e742863fdbe2","steps":["basis A013611"],"edges":["e-7b93be005ec5b750"],"text":"A038305 \u2192 basis A013611"},"headline":"A038305 \u2192 basis A013611","chain":null}},{"id":"l-88cb1c8cac5d8324","kind":"fingerprint_related","node_a_id":"n-fbee24f9afac6b9c","node_b_id":"n-5327121044655cf2","fingerprint":"f6-42b2d489dcb16cdd","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-42b2d489dcb16cdd"},"other_id":"n-5327121044655cf2","other":{"id":"n-5327121044655cf2","name":"basis_map_A013609 of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*6^j.","first_index":0,"fingerprint":"f6-42b2d489dcb16cdd","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["676","1248","576"],["17576","48672","44928","13824"],["456976","1687296","2336256","1437696","331776"]],"card_rows":[["676","1248","576"],["17576","48672","44928","13824"],["456976","1687296","2336256","1437696","331776"],["11881376","54837120","101237760","93450240","43130880","7962624"]],"row_sums_preview":["2500","125000","6250000","312500000","15625000000","781250000000","39062500000000","1953125000000000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c11cb6bca0e3e7dc","steps":["basis A013609","basis A013609"],"edges":["e-65e466871b01ed5e","e-d912b514a96be6b4"],"text":"A038284 \u2192 basis A013609 \u2192 basis A013609"},"headline":"A038284 \u2192 basis A013609 \u2192 basis A013609","chain":null}},{"id":"l-8c6b785bb5d23023","kind":"fingerprint_related","node_a_id":"n-6e81fa7b0bac1617","node_b_id":"n-fbee24f9afac6b9c","fingerprint":"f6-42b2d489dcb16cdd","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-42b2d489dcb16cdd"},"other_id":"n-6e81fa7b0bac1617","other":{"id":"n-6e81fa7b0bac1617","name":"mobius_x_over_x_plus_one of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*1^j.","first_index":0,"fingerprint":"f6-42b2d489dcb16cdd","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["12","13"],["144","312","169"]],"card_rows":[["1"],["12","13"],["144","312","169"],["1728","5616","6084","2197"]],"row_sums_preview":["1","25","625","15625","390625","9765625","244140625","6103515625"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-81e6fb4f9f226d77","steps":["x \u21a6 x/(x+1)"],"edges":["e-5c0a38aee3aef05d"],"text":"A038327 \u2192 x \u21a6 x/(x+1)"},"headline":"A038327 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-9c232e68d555660d","kind":"fingerprint_related","node_a_id":"n-c39ad02ee0de214a","node_b_id":"n-fbee24f9afac6b9c","fingerprint":"f6-42b2d489dcb16cdd","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-42b2d489dcb16cdd"},"other_id":"n-c39ad02ee0de214a","other":{"id":"n-c39ad02ee0de214a","name":"reversal of basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*3^j.","first_index":0,"fingerprint":"f6-42b2d489dcb16cdd","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["144","312","169"],["1728","5616","6084","2197"],["20736","89856","146016","105456","28561"]],"card_rows":[["144","312","169"],["1728","5616","6084","2197"],["20736","89856","146016","105456","28561"],["248832","1347840","2920320","3163680","1713660","371293"]],"row_sums_preview":["625","15625","390625","9765625","244140625","6103515625","152587890625","3814697265625"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-4397e742863fdbe2","steps":["basis A013611","reverse"],"edges":["e-7b93be005ec5b750","e-2477911ee3e8d98b"],"text":"A038305 \u2192 basis A013611 \u2192 reverse"},"headline":"A038305 \u2192 basis A013611 \u2192 reverse","chain":null}},{"id":"l-ec43f993150b3c2f","kind":"fingerprint_related","node_a_id":"n-bb55a9508cc434b3","node_b_id":"n-fbee24f9afac6b9c","fingerprint":"f6-42b2d489dcb16cdd","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-42b2d489dcb16cdd"},"other_id":"n-bb55a9508cc434b3","other":{"id":"n-bb55a9508cc434b3","name":"derivative of basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*3^j.","first_index":0,"fingerprint":"f6-42b2d489dcb16cdd","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["312","288"],["6084","11232","5184"],["105456","292032","269568","82944"]],"card_rows":[["312","288"],["6084","11232","5184"],["105456","292032","269568","82944"],["1713660","6327360","8760960","5391360","1244160"]],"row_sums_preview":["600","22500","750000","23437500","703125000","20507812500","585937500000","16479492187500"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-4397e742863fdbe2","steps":["basis A013611","D"],"edges":["e-7b93be005ec5b750","e-2f410ecb54debb66"],"text":"A038305 \u2192 basis A013611 \u2192 D"},"headline":"A038305 \u2192 basis A013611 \u2192 D","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:           1;\n  1:          13,          12;\n  2:         169,         312,          144;\n  3:        2197,        6084,         5616,         1728;\n  4:       28561,      105456,       146016,        89856,        20736;\n  5:      371293,     1713660,      3163680,      2920320,      1347840,       248832;\n  6:     4826809,    26733096,     61691760,     75928320,     52565760,     19408896,      2985984;\n  7:    62748517,   405451956,   1122790032,   1727369280,   1594494720,    883104768,    271724544,     35831808;\n  8:   815730721,  6023857632,  19461693888,  35929281024,  41456862720,  30614298624,  14129676288,   3726508032,   429981696;\n  9: 10604499373, 88098917868, 325288312128, 700620979968, 970090587648, 895468234752, 551057375232, 218000719872, 50307858432, 5159780352;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (13+12*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(13 + 12*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (13 + 12*x)*v[i - 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-366b2525683554ec","steps":["basis A013611","basis A013610"],"edges":["e-b30dba0e9f8b339c","e-1e50496cc65ef5b9"],"text":"A038279 \u2192 basis A013611 \u2192 basis A013610"},"headline":"A038279 \u2192 basis A013611 \u2192 basis A013610","chain":null}