{"id":"n-71cef8c9aa2e921b","revision":1,"first_index":1,"fingerprint":"f6-efbf5758f6fa5da7","name":"obreschkoff_combination of Triangle read by rows: 2-Stirling numbers of the second kind.","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":false,"rows":[["3","1"],["6","6","1"],["12","24","10","1"],["24","84","64","15","1"],["48","276","340","139","21","1"],["96","876","1636","1035","265","28","1"],["192","2724","7420","6811","2625","461","36","1"],["384","8364","32404","41475","22561","5852","749","45","1"],["768","25476","137980","239779","176841","63525","11844","1154","55","1"],["1536","77196","577396","1336875","1300825","621516","158277","22230","1704","66","1"],["3072","233124","2386780","7261771","9141825","5651437","1887732","358347","39270","2430","78","1"],["6144","702444","9780244","38695635","62112721","48701884","20753293","5112855","751047","66000","3366","91","1"],["12288","2113476","39823420","203258419","411371961","403025909","214728228","66768988","12623325","1477047","106392","4549","105","1"],["24576","6352716","161407156","1056115515","2671490185","3232553324","2120851733","815649120","193002238","28870842","2753751","165529","6019","120","1"],["49152","19082724","651981340","5441984731","17085056625","25299363453","20199367188","9461693813","2745671500","510581500","61915854","4905628","249795","7819","136","1"]],"degrees":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],"row_sums":["4","13","47","188","825","3937","20270","111835","657423","4097622","26965867","186685725","1355314108","10289242825","81481911259"],"nonn":true,"definitions":[{"position":0,"source":"derived:obreschkoff_combination","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["3","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["2","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":162727,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, +inf)","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)","region_exact":null,"edges":["e-f5ad2d4d1fdd1ba8"]}],"region_exact":{},"incoming_edges":[{"id":"e-f5ad2d4d1fdd1ba8","operation":{"operation":"obreschkoff_combination"},"operation_detail":null,"conclusion_id":"n-71cef8c9aa2e921b","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":"91bea50c145d5f8a4e374473355d740ac475a97b6a324970c4056d3c8bb32ac7","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{}}],"premises":[{"position":0,"node_id":"n-6aaec68d653cbdb0","role":"sequence","name":"Triangle read by rows: 2-Stirling numbers of the second kind.","uses":{"claim":"real_rooted","from_index":1,"region":{}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)"},{"position":1,"node_id":"n-6aaec68d653cbdb0","role":"sequence","name":"Triangle read by rows: 2-Stirling numbers of the second kind.","uses":{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"},"uses_text":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-c0c7074bb89eafdb","kind":"oeis_match","node_a_id":"n-71cef8c9aa2e921b","node_b_id":null,"fingerprint":null,"oeis_id":"A078468","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":1,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A078468","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A078468","relation":"match_row_sums"}],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:    3,     1;\n   2:    6,     6,      1;\n   3:   12,    24,     10,       1;\n   4:   24,    84,     64,      15,       1;\n   5:   48,   276,    340,     139,      21,      1;\n   6:   96,   876,   1636,    1035,     265,     28,      1;\n   7:  192,  2724,   7420,    6811,    2625,    461,     36,     1;\n   8:  384,  8364,  32404,   41475,   22561,   5852,    749,    45,    1;\n   9:  768, 25476, 137980,  239779,  176841,  63525,  11844,  1154,   55,  1;\n  10: 1536, 77196, 577396, 1336875, 1300825, 621516, 158277, 22230, 1704, 66, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2+x)*P(n-1,x) + x*d/dx P(n-1,x) for n >= 2, with P(1,x) = 3 + x.","mathematica":"P[1, x_] := 3 + x;\nP[n_, x_] := P[n, x] = Expand[(2 + x)*P[n - 1, x] + x*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 3 + x; for(n = 2, N, my(i = n); v[i] = (2 + x)*v[i - 1] + x*deriv(v[i - 1], x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-6aaec68d653cbdb0","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-f5ad2d4d1fdd1ba8"],"text":"A143494 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A143494 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}
