{"id":"n-37c70c80fa31c1b5","revision":1,"first_index":0,"fingerprint":"f6-ee779bc3aafd2f30","name":"obreschkoff_combination 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":1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[["11","1"],["110","21","1"],["1100","320","31","1"],["11000","4300","630","41","1"],["110000","54000","10600","1040","51","1"],["1100000","650000","160000","21000","1550","61","1"],["11000000","7600000","2250000","370000","36500","2160","71","1"],["110000000","87000000","30100000","5950000","735000","58100","2870","81","1"],["1100000000","980000000","388000000","89600000","13300000","1316000","86800","3680","91","1"],["11000000000","10900000000","4860000000","1284000000","222600000","26460000","2184000","123600","4590","101","1"],["110000000000","120000000000","59500000000","17700000000","3510000000","487200000","48300000","3420000","169500","5600","111","1"],["1100000000000","1310000000000","715000000000","236500000000","52800000000","8382000000","970200000","82500000","5115000","225500","6710","121","1"],["11000000000000","14200000000000","8460000000000","3080000000000","764500000000","136620000000","18084000000","1795200000","133650000","7370000","292600","7920","131","1"],["110000000000000","153000000000000","98800000000000","39260000000000","10725000000000","2130700000000","317460000000","36036000000","3131700000","207350000","10296000","371800","9230","141","1"],["1100000000000000","1640000000000000","1141000000000000","491400000000000","146510000000000","32032000000000","5305300000000","677820000000","67353000000","5205200000","310310000","14014000","464100","10640","151","1"]],"degrees":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],"row_sums":["12","132","1452","15972","175692","1932612","21258732","233846052","2572306572","28295372292","311249095212","3423740047332","37661140520652","414272545727172","4556998002998892"],"nonn":true,"definitions":[{"position":0,"source":"derived:obreschkoff_combination","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["11","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["10","1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":162822,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, +inf)","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)","region_exact":null,"edges":["e-e361768fdb02a8d8"]}],"region_exact":{},"incoming_edges":[{"id":"e-e361768fdb02a8d8","operation":{"operation":"obreschkoff_combination"},"operation_detail":null,"conclusion_id":"n-37c70c80fa31c1b5","premise_count":2,"theorem":"Hermite--Kakeya--Obreschkoff: real-rooted f, g with interlacing zeros give a zero or real-rooted a f + b g for all real a, b (Obreschkoff 1963; Fisk, Polynomials, roots, and interlacing, Thm 1.43)","lean_route":null,"lean_declarations":[],"certificate_hash":"e5eecccf0815b9099c6de89317a6ac054a5ab069fb81a9381ba141a4a0186374","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"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":{}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)"},{"position":1,"node_id":"n-a920b56c9b3ef4d1","role":"sequence","name":"Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*1^j.","uses":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"uses_text":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-0ccb1d95798e87d4","kind":"oeis_match","node_a_id":"n-37c70c80fa31c1b5","node_b_id":null,"fingerprint":null,"oeis_id":"A168929","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":1,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A168929","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-2720e8cc5fcf49d4","kind":"oeis_match","node_a_id":"n-37c70c80fa31c1b5","node_b_id":null,"fingerprint":null,"oeis_id":"A168785","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":1,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A168785","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-3e28da6ab5a1abf0","kind":"oeis_match","node_a_id":"n-37c70c80fa31c1b5","node_b_id":null,"fingerprint":null,"oeis_id":"A003954","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":1,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A003954","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-5360a83bbbbabe7f","kind":"oeis_match","node_a_id":"n-37c70c80fa31c1b5","node_b_id":null,"fingerprint":null,"oeis_id":"A168737","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":1,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A168737","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-56419e7ef50a08fb","kind":"oeis_match","node_a_id":"n-37c70c80fa31c1b5","node_b_id":null,"fingerprint":null,"oeis_id":"A168881","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":1,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A168881","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-c1bf95f5dac7c7c4","kind":"oeis_match","node_a_id":"n-37c70c80fa31c1b5","node_b_id":null,"fingerprint":null,"oeis_id":"A168833","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":1,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A168833","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-ca255eab67faead1","kind":"oeis_match","node_a_id":"n-37c70c80fa31c1b5","node_b_id":null,"fingerprint":null,"oeis_id":"A168689","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":1,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A168689","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-d1dd98101c05e34b","kind":"oeis_match","node_a_id":"n-37c70c80fa31c1b5","node_b_id":null,"fingerprint":null,"oeis_id":"A167916","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":1,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A167916","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A003954","relation":"match_row_sums"},{"oeis_id":"A167916","relation":"match_row_sums"},{"oeis_id":"A168689","relation":"match_row_sums"},{"oeis_id":"A168737","relation":"match_row_sums"},{"oeis_id":"A168785","relation":"match_row_sums"},{"oeis_id":"A168833","relation":"match_row_sums"},{"oeis_id":"A168881","relation":"match_row_sums"},{"oeis_id":"A168929","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:          11,           1;\n  1:         110,          21,          1;\n  2:        1100,         320,         31,          1;\n  3:       11000,        4300,        630,         41,         1;\n  4:      110000,       54000,      10600,       1040,        51,        1;\n  5:     1100000,      650000,     160000,      21000,      1550,       61,       1;\n  6:    11000000,     7600000,    2250000,     370000,     36500,     2160,      71,      1;\n  7:   110000000,    87000000,   30100000,    5950000,    735000,    58100,    2870,     81,    1;\n  8:  1100000000,   980000000,  388000000,   89600000,  13300000,  1316000,   86800,   3680,   91,   1;\n  9: 11000000000, 10900000000, 4860000000, 1284000000, 222600000, 26460000, 2184000, 123600, 4590, 101, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (10+x)*P(n-1,x) for n >= 1, with P(0,x) = 11 + x.","mathematica":"P[0, x_] := 11 + x;\nP[n_, x_] := P[n, x] = Expand[(10 + 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] = 11 + x; for(n = 1, N, my(i = n + 1); v[i] = (10 + 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":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-e361768fdb02a8d8"],"text":"A038303 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A038303 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}