{"id":"n-10f70e3ea8b205cf","revision":1,"first_index":0,"fingerprint":"f6-9bb94729b5d21425","name":"basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*1^j.","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"],["11","2"],["121","44","4"],["1331","726","132","8"],["14641","10648","2904","352","16"],["161051","146410","53240","9680","880","32"],["1771561","1932612","878460","212960","29040","2112","64"],["19487171","24801854","13528284","4099480","745360","81312","4928","128"],["214358881","311794736","198414832","72150848","16397920","2385152","216832","11264","256"],["2357947691","3858459858","2806152624","1190488992","324678816","59032512","7155456","557568","25344","512"],["25937424601","47158953820","38584598580","18707684160","5952444960","1298715264","196775040","20444160","1393920","56320","1024"],["285311670611","570623341222","518748492020","282953722920","102892262880","26190757824","4761955968","618435840","56221440","3407360","123904","2048"],["3138428376721","6847480094664","6847480094664","4149987936160","1697722337520","493882861824","104763031296","16326706176","1855307520","149923840","8177664","270336","4096"],["34522712143931","81599137794746","89017241230632","59344827487088","26974921585040","8828156155104","2140159067904","389119830528","53061795072","5359777280","389801984","19329024","585728","8192"],["379749833583241","966635940030068","1142387929126444","830827584819232","415413792409616","151059560876224","41198062057152","8560636271616","1361919406848","165081140224","15007376384","992223232","45101056","1261568","16384"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","13","169","2197","28561","371293","4826809","62748517","815730721","10604499373","137858491849","1792160394037","23298085122481","302875106592253","3937376385699289"],"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":[["11","2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":166801,"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-71aa91d8d70f875b"]},{"id":166802,"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-71aa91d8d70f875b"]},{"id":166803,"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-71aa91d8d70f875b"]}],"region_exact":{},"incoming_edges":[{"id":"e-71aa91d8d70f875b","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-10f70e3ea8b205cf","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":"aca02e90f52afbb5008d95dfdfbad6726044f5719add87995fa3e37d5349f9fb","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-a920b56c9b3ef4d1","role":"sequence","name":"Triangle whose (i,j)-th entry is binomial(i,j)*10^(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-224787304d3f87f6","kind":"oeis_match","node_a_id":"n-10f70e3ea8b205cf","node_b_id":null,"fingerprint":null,"oeis_id":"A001022","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A001022","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-7e56bf645ba24ae4","kind":"oeis_match","node_a_id":"n-10f70e3ea8b205cf","node_b_id":null,"fingerprint":null,"oeis_id":"A038316","in_catalog":0,"target":"triangle","row_alignment":"tabl","rows_matched":8,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":8,"target":"triangle","terms_compared":37},"in_catalog":false,"oeis_id":"A038316","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-af8ca049cf6876ea","kind":"oeis_match","node_a_id":"n-10f70e3ea8b205cf","node_b_id":null,"fingerprint":null,"oeis_id":"A038217","in_catalog":0,"target":"reversed_triangle","row_alignment":"tabl","rows_matched":8,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":8,"target":"reversed_triangle","terms_compared":38},"in_catalog":false,"oeis_id":"A038217","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A038217","relation":"match_reversed_triangle"},{"oeis_id":"A001022","relation":"match_row_sums"},{"oeis_id":"A038316","relation":"match_triangle"}],"bfiles":[{"node_id":"n-10f70e3ea8b205cf","oeis_id":"A001022","target":"row_sums","oeis_offset":0,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":21,"oeis_terms_available":21,"term_count":200,"raw_size":23159},{"node_id":"n-10f70e3ea8b205cf","oeis_id":"A038217","target":"reversed_triangle","oeis_offset":0,"first_row":0,"rows_used":44,"terms_matched":38,"oeis_terms_agreeing":38,"oeis_terms_available":38,"term_count":990,"raw_size":29676},{"node_id":"n-10f70e3ea8b205cf","oeis_id":"A038316","target":"triangle","oeis_offset":0,"first_row":0,"rows_used":44,"terms_matched":37,"oeis_terms_agreeing":37,"oeis_terms_available":37,"term_count":990,"raw_size":29676}],"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:         11,          2;\n  2:        121,         44,          4;\n  3:       1331,        726,        132,          8;\n  4:      14641,      10648,       2904,        352,        16;\n  5:     161051,     146410,      53240,       9680,       880,       32;\n  6:    1771561,    1932612,     878460,     212960,     29040,     2112,      64;\n  7:   19487171,   24801854,   13528284,    4099480,    745360,    81312,    4928,    128;\n  8:  214358881,  311794736,  198414832,   72150848,  16397920,  2385152,  216832,  11264,   256;\n  9: 2357947691, 3858459858, 2806152624, 1190488992, 324678816, 59032512, 7155456, 557568, 25344, 512;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (11+2*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[(11 + 2*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] = (11 + 2*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}},{"oeis_id":"A038217","target":"reversed_triangle","first_row":0,"oeis_offset":0,"reversed":true,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:   1;\n  1:   2,    11;\n  2:   4,    44,    121;\n  3:   8,   132,    726,    1331;\n  4:  16,   352,   2904,   10648,    14641;\n  5:  32,   880,   9680,   53240,   146410,    161051;\n  6:  64,  2112,  29040,  212960,   878460,   1932612,    1771561;\n  7: 128,  4928,  81312,  745360,  4099480,  13528284,   24801854,   19487171;\n  8: 256, 11264, 216832, 2385152, 16397920,  72150848,  198414832,  311794736,  214358881;\n  9: 512, 25344, 557568, 7155456, 59032512, 324678816, 1190488992, 2806152624, 3858459858, 2357947691;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (11+2*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[(11 + 2*x)*P[n - 1, x]];\nTable[Reverse[PadRight[CoefficientList[P[n, x], x], n + 1]], {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] = (11 + 2*x)*v[i - 1]); v};\nrow(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c = Vecrev(c); 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":true,"trim_zeros":"none","regular_triangle":true}},{"oeis_id":"A038316","target":"triangle","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:          1;\n  1:         11,          2;\n  2:        121,         44,          4;\n  3:       1331,        726,        132,          8;\n  4:      14641,      10648,       2904,        352,        16;\n  5:     161051,     146410,      53240,       9680,       880,       32;\n  6:    1771561,    1932612,     878460,     212960,     29040,     2112,      64;\n  7:   19487171,   24801854,   13528284,    4099480,    745360,    81312,    4928,    128;\n  8:  214358881,  311794736,  198414832,   72150848,  16397920,  2385152,  216832,  11264,   256;\n  9: 2357947691, 3858459858, 2806152624, 1190488992, 324678816, 59032512, 7155456, 557568, 25344, 512;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (11+2*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[(11 + 2*x)*P[n - 1, x]];\nTable[PadRight[CoefficientList[P[n, x], x], n + 1], {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] = (11 + 2*x)*v[i - 1]); v};\nrow(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); 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":true}}],"derivation":{"root":"n-a920b56c9b3ef4d1","steps":["basis A013609"],"edges":["e-71aa91d8d70f875b"],"text":"A038303 \u2192 basis A013609"},"headline":"A038303 \u2192 basis A013609","chain":null}
