{"id":"n-f562ab3e3b669a1b","revision":1,"first_index":0,"fingerprint":"f6-c62c9860d860e562","name":"basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*6^(i-j)*5^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","15"],["121","330","225"],["1331","5445","7425","3375"],["14641","79860","163350","148500","50625"],["161051","1098075","2994750","4083750","2784375","759375"],["1771561","14494590","49413375","89842500","91884375","50118750","11390625"],["19487171","186013905","760965975","1729468125","2358365625","1929571875","877078125","170859375"],["214358881","2338460520","11160834300","30438639000","51884043750","56600775000","38591437500","15035625000","2562890625"],["2357947691","28938448935","157846085100","502237543500","1027304066250","1400869181250","1273517437500","744263437500","253726171875","38443359375"],["25937424601","353692153650","2170383670125","7892304255000","18833907881250","30819121987500","35021729531250","27289659375000","13954939453125","4228769531250","576650390625"],["285311670611","4279675059165","29179602676125","119371101856875","325557550518750","621518960081250","847525854656250","825512196093750","562849224609375","255840556640625","69774697265625","8649755859375"],["3138428376721","51356100709980","385170755324850","1750776160567500","5371699583559375","11720071818675000","18645568802437500","21793521976875000","18574024412109375","11256984492187500","4605130019531250","1141767773437500","129746337890625"],["34522712143931","611993533460595","5007219819223050","25036099096115250","85350337827665625","209496283758815625","380902334106937500","519412273782187500","531217098186328125","402437195595703125","219511197597656250","81636395800781250","18553726318359375","1946195068359375"],["379749833583241","7249769550225510","64259321013362475","350505387345613500","1314395202546050625","3584714188761956250","7332369931558546875","11427070023208125000","13634572186782421875","12395065624347656250","8451181107509765625","4190668317773437500","1428636926513671875","299714040527343750","29192926025390625"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","26","676","17576","456976","11881376","308915776","8031810176","208827064576","5429503678976","141167095653376","3670344486987776","95428956661682176","2481152873203736576","64509974703297150976"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013610","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["11","15"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":167475,"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-aef71e51dba5316b"]},{"id":167476,"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-aef71e51dba5316b"]},{"id":167477,"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-aef71e51dba5316b"]}],"region_exact":{},"incoming_edges":[{"id":"e-aef71e51dba5316b","operation":{"map":"A013610","operation":"basis_map"},"operation_detail":"A013610","conclusion_id":"n-f562ab3e3b669a1b","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":"0daf8231ad587bb6bad495ca49a342d6c3b8672268aa974e8b1491b1c2784093","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-e6ebe74518d42cbe","role":"sequence","name":"Triangle whose (i,j)-th entry is binomial(i,j)*6^(i-j)*5^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-4e411db0c8c498fb","kind":"fingerprint_related","node_a_id":"n-f562ab3e3b669a1b","node_b_id":"n-058123003a903c80","fingerprint":"f6-c62c9860d860e562","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-c62c9860d860e562"},"other_id":"n-058123003a903c80","other":{"id":"n-058123003a903c80","name":"basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*10^j.","first_index":0,"fingerprint":"f6-c62c9860d860e562","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["484","1320","900"],["10648","43560","59400","27000"],["234256","1277760","2613600","2376000","810000"]],"card_rows":[["484","1320","900"],["10648","43560","59400","27000"],["234256","1277760","2613600","2376000","810000"],["5153632","35138400","95832000","130680000","89100000","24300000"]],"row_sums_preview":["2704","140608","7311616","380204032","19770609664","1028071702528","53459728531456","2779905883635712"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ec89fb3fc20684fc","steps":["basis A013610"],"edges":["e-022a6c0efe3a8d25"],"text":"A038336 \u2192 basis A013610"},"headline":"A038336 \u2192 basis A013610","chain":null}},{"id":"l-5d787649ed6c720e","kind":"oeis_match","node_a_id":"n-f562ab3e3b669a1b","node_b_id":null,"fingerprint":null,"oeis_id":"A009970","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":"A009970","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A009970","relation":"match_row_sums"}],"bfiles":[{"node_id":"n-f562ab3e3b669a1b","oeis_id":"A009970","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":29149}],"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,          15;\n  2:        121,         330,          225;\n  3:       1331,        5445,         7425,         3375;\n  4:      14641,       79860,       163350,       148500,         50625;\n  5:     161051,     1098075,      2994750,      4083750,       2784375,        759375;\n  6:    1771561,    14494590,     49413375,     89842500,      91884375,      50118750,      11390625;\n  7:   19487171,   186013905,    760965975,   1729468125,    2358365625,    1929571875,     877078125,    170859375;\n  8:  214358881,  2338460520,  11160834300,  30438639000,   51884043750,   56600775000,   38591437500,  15035625000,   2562890625;\n  9: 2357947691, 28938448935, 157846085100, 502237543500, 1027304066250, 1400869181250, 1273517437500, 744263437500, 253726171875, 38443359375;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (11+15*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 + 15*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 + 15*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-e6ebe74518d42cbe","steps":["basis A013610"],"edges":["e-aef71e51dba5316b"],"text":"A038259 \u2192 basis A013610"},"headline":"A038259 \u2192 basis A013610","chain":null}
