{"id":"n-a875e5d4789de94f","revision":1,"first_index":0,"fingerprint":"f6-d935ec63feee39a0","name":"basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*5^j.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":2,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[["256","480","225"],["4096","11520","10800","3375"],["65536","245760","345600","216000","50625"],["1048576","4915200","9216000","8640000","4050000","759375"],["16777216","94371840","221184000","276480000","194400000","72900000","11390625"],["268435456","1761607680","4954521600","7741440000","7257600000","4082400000","1275750000","170859375"],["4294967296","32212254720","105696460800","198180864000","232243200000","174182400000","81648000000","21870000000","2562890625"],["68719476736","579820584960","2174327193600","4756340736000","6688604160000","6270566400000","3919104000000","1574640000000","369056250000","38443359375"],["1099511627776","10307921510400","43486543872000","108716359680000","178362777600000","200658124800000","156764160000000","83980800000000","29524500000000","6150937500000","576650390625"],["17592186044416","181419418583040","850403524608000","2391759912960000","4484549836800000","5885971660800000","5518098432000000","3695155200000000","1732104000000000","541282500000000","101490468750000","8649755859375"],["281474976710656","3166593487994880","16327747672473600","51024211476480000","107629196083200000","161443794124800000","176579149824000000","141893959680000000","83140992000000000","34642080000000000","9743085000000000","1660753125000000","129746337890625"],["4503599627370496","54887620458577920","308742865079500800","1061303598710784000","2487430309478400000","4197538647244800000","5246923309056000000","4918990602240000000","3458665267200000000","1801388160000000000","675520560000000000","172718325000000000","26987238281250000","1946195068359375"],["72057594037927936","945755921747804160","5763200148150681600","21612000555565056000","55718438932316160000","104472072998092800000","146913852653568000000","157407699271680000000","129123503308800000000","80702189568000000000","37829151360000000000","12896301600000000000","3022570687500000000","435947695312500000","29192926025390625"],["1152921504606846976","16212958658533785600","106397541196627968000","432240011111301120000","1215675031250534400000","2507329751954227200000","3917702737428480000000","4722230978150400000000","4427091542016000000000","3228087582720000000000","1815799265280000000000","773778096000000000000","241805655000000000000","52313723437500000000","7006302246093750000","437893890380859375"],["18446744073709551616","276701161105643274240","1945555039024054272000","8511803295730237440000","25934400666678067200000","58352401500025651200000","100293190078169088000000","134321236711833600000000","141666929344512000000000","118055774453760000000000","77474101985280000000000","39617438515200000000000","15475561920000000000000","4464104400000000000000","896806687500000000000","112100835937500000000","6568408355712890625"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["961","29791","923521","28629151","887503681","27512614111","852891037441","26439622160671","819628286980801","25408476896404831","787662783788549761","24417546297445042591","756943935220796320321","23465261991844685929951","727423121747185263828481"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013610","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["256","480","225"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["16","15"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":167573,"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-fd6222a072627022"]},{"id":167574,"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-fd6222a072627022"]},{"id":167575,"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-fd6222a072627022"]}],"region_exact":{},"incoming_edges":[{"id":"e-fd6222a072627022","operation":{"map":"A013610","operation":"basis_map"},"operation_detail":"A013610","conclusion_id":"n-a875e5d4789de94f","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":"43cc6e279ebf20948caf93a1364dacbdce716f0e9a5c744a2871afb662de92a9","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-2bc1923905556bf6","role":"sequence","name":"Triangle whose (i,j)-th entry is binomial(i,j)*11^(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-540b059701de35e0","kind":"oeis_match","node_a_id":"n-a875e5d4789de94f","node_b_id":null,"fingerprint":null,"oeis_id":"A009975","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":2,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A009975","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A009975","relation":"match_row_sums"}],"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:            256,             480,             225;\n  1:           4096,           11520,           10800,             3375;\n  2:          65536,          245760,          345600,           216000,            50625;\n  3:        1048576,         4915200,         9216000,          8640000,          4050000,           759375;\n  4:       16777216,        94371840,       221184000,        276480000,        194400000,         72900000,         11390625;\n  5:      268435456,      1761607680,      4954521600,       7741440000,       7257600000,       4082400000,       1275750000,        170859375;\n  6:     4294967296,     32212254720,    105696460800,     198180864000,     232243200000,     174182400000,      81648000000,      21870000000,       2562890625;\n  7:    68719476736,    579820584960,   2174327193600,    4756340736000,    6688604160000,    6270566400000,    3919104000000,    1574640000000,     369056250000,     38443359375;\n  8:  1099511627776,  10307921510400,  43486543872000,  108716359680000,  178362777600000,  200658124800000,  156764160000000,   83980800000000,   29524500000000,   6150937500000,    576650390625;\n  9: 17592186044416, 181419418583040, 850403524608000, 2391759912960000, 4484549836800000, 5885971660800000, 5518098432000000, 3695155200000000, 1732104000000000, 541282500000000, 101490468750000, 8649755859375;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (16+15*x)*P(n-1,x) for n >= 1, with P(0,x) = 256 + 480*x + 225*x^2.","mathematica":"P[0, x_] := 256 + 480*x + 225*x^2;\nP[n_, x_] := P[n, x] = Expand[(16 + 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] = 256 + 480*x + 225*x^2; for(n = 1, N, my(i = n + 1); v[i] = (16 + 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-2bc1923905556bf6","steps":["basis A013610"],"edges":["e-fd6222a072627022"],"text":"A038319 \u2192 basis A013610"},"headline":"A038319 \u2192 basis A013610","chain":null}