{"id":"n-f5a3fd4262a598df","revision":0,"first_index":0,"fingerprint":"f6-9bb94729b5d21425","name":"reversal of 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"],["2","11"],["4","44","121"],["8","132","726","1331"],["16","352","2904","10648","14641"],["32","880","9680","53240","146410","161051"],["64","2112","29040","212960","878460","1932612","1771561"],["128","4928","81312","745360","4099480","13528284","24801854","19487171"],["256","11264","216832","2385152","16397920","72150848","198414832","311794736","214358881"],["512","25344","557568","7155456","59032512","324678816","1190488992","2806152624","3858459858","2357947691"],["1024","56320","1393920","20444160","196775040","1298715264","5952444960","18707684160","38584598580","47158953820","25937424601"],["2048","123904","3407360","56221440","618435840","4761955968","26190757824","102892262880","282953722920","518748492020","570623341222","285311670611"],["4096","270336","8177664","149923840","1855307520","16326706176","104763031296","493882861824","1697722337520","4149987936160","6847480094664","6847480094664","3138428376721"],["8192","585728","19329024","389801984","5359777280","53061795072","389119830528","2140159067904","8828156155104","26974921585040","59344827487088","89017241230632","81599137794746","34522712143931"],["16384","1261568","45101056","992223232","15007376384","165081140224","1361919406848","8560636271616","41198062057152","151059560876224","415413792409616","830827584819232","1142387929126444","966635940030068","379749833583241"]],"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:reversal","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["2","11"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":187789,"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-bc8497f8149bdee2"]},{"id":187790,"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-bc8497f8149bdee2"]},{"id":187791,"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-bc8497f8149bdee2"]}],"region_exact":{},"incoming_edges":[{"id":"e-bc8497f8149bdee2","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"reversal\",\"schedule\":{\"first_index\":0,\"first_degree\":0,\"step\":1}}"},"operation_detail":"reversal, shift 0","conclusion_id":"n-f5a3fd4262a598df","premise_count":1,"theorem":"a real affine substitution x -> a x + b (a != 0), the reversal x^d p(1/x) (d >= deg p) and a real Mobius map (c x + e)^d p((a x + b)/(c x + e)) (a e - b c != 0, d >= deg p) map real roots to real roots: a root r of p gives the root (r - b)/a, 1/r (and 0) or (e r - b)/(a - c r) (and -e/c); for an input row that is a polynomial the degree condition holds automatically","lean_route":"reversed","lean_declarations":["RealRooted.X_pow_mul_reverse_eq_zero_or_splits","RealRooted.reflect_eq_zero_or_splits"],"certificate_hash":"5632e77855070bf584cc13f14fa98acc8f08c6fa543169afb61d1b1a1b5ed58b","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-10f70e3ea8b205cf","role":"sequence","name":"basis_map_A013609 of 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-9c27a091cd5971f7","kind":"fingerprint_related","node_a_id":"n-6d4094237cfa9611","node_b_id":"n-f5a3fd4262a598df","fingerprint":"f6-9bb94729b5d21425","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-9bb94729b5d21425"},"other_id":"n-6d4094237cfa9611","other":{"id":"n-6d4094237cfa9611","name":"derivative of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*1^j.","first_index":0,"fingerprint":"f6-9bb94729b5d21425","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["2"],["44","8"]],"card_rows":[["2"],["44","8"],["726","264","24"],["10648","5808","1056","64"]],"row_sums_preview":["0","2","52","1014","17576","285610","4455516","67575326"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a920b56c9b3ef4d1","steps":["basis A013609","D"],"edges":["e-71aa91d8d70f875b","e-6baf1dc14d8667a7"],"text":"A038303 \u2192 basis A013609 \u2192 D"},"headline":"A038303 \u2192 basis A013609 \u2192 D","chain":null}},{"id":"l-da8d2ebd5f0b7713","kind":"fingerprint_related","node_a_id":"n-10f70e3ea8b205cf","node_b_id":"n-f5a3fd4262a598df","fingerprint":"f6-9bb94729b5d21425","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-9bb94729b5d21425"},"other_id":"n-10f70e3ea8b205cf","other":{"id":"n-10f70e3ea8b205cf","name":"basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*1^j.","first_index":0,"fingerprint":"f6-9bb94729b5d21425","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["11","2"],["121","44","4"]],"card_rows":[["1"],["11","2"],["121","44","4"],["1331","726","132","8"]],"row_sums_preview":["1","13","169","2197","28561","371293","4826809","62748517"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a920b56c9b3ef4d1","steps":["basis A013609"],"edges":["e-71aa91d8d70f875b"],"text":"A038303 \u2192 basis A013609"},"headline":"A038303 \u2192 basis A013609","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:   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":"The row polynomials satisfy P(n,x) = (2+11*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[(2 + 11*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] = (2 + 11*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-a920b56c9b3ef4d1","steps":["basis A013609","reverse"],"edges":["e-71aa91d8d70f875b","e-bc8497f8149bdee2"],"text":"A038303 \u2192 basis A013609 \u2192 reverse"},"headline":"A038303 \u2192 basis A013609 \u2192 reverse","chain":null}
