{"id":"n-a712a218f238860b","revision":1,"first_index":1,"fingerprint":"f6-bf3526b154ae1214","name":"obreschkoff_combination of Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by the n-th Catalan number (A000108).","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":[["2","1"],["3","5","2"],["7","19","17","5"],["19","71","99","61","14"],["56","266","504","476","224","42"],["174","1002","2400","3060","2190","834","132"],["561","3795","10989","17655","16995","9801","3135","429"],["1859","14443","49049","95095","115115","89089","43043","11869","1430"],["6292","55198","215072","488488","712712","692692","448448","186472","45188","4862"],["21658","211718","930852","2423928","4139772","4845204","3935568","2190552","799578","172822","16796"],["75582","814606","3989050","11715210","22926540","31391724","30686292","21414900","10455510","3401190","663442","58786"],["266798","3142790","16962022","55462330","122365320","191904636","219362220","184144884","112665630","48995870","14375438","2554930","208012"],["950912","12153844","71674992","258232040","634139440","1120857804","1467019488","1439561904","1059078240","576936140","226198192","60442344","9865712","742900"],["3417340","47099860","301320240","1185965560","3208287940","6310341180","9306159720","10453494480","8987455620","5885402380","2889583840","1031442360","253031740","38185060","2674440"],["12369285","182864835","1261332765","5384650635","15910577865","34467848415","56554042545","71565005655","70417670895","53876928105","31790733975","14206959585","4654528515","1055400885","148097115","9694845"]],"degrees":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],"row_sums":["3","10","48","264","1568","9792","63360","420992","2855424","19688448","137592832","972410880","6937853952","49903861760","361498705920"],"nonn":true,"definitions":[{"position":0,"source":"derived:obreschkoff_combination","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["2","1"]],"recurrence":{"denominator":{"coeffs":[["-6","-6"],["-1","-2"],["5","4"]]},"terms":[{"coeff":{"coeffs":[["6","18","12"],["-34","-66","-32"],["20","36","16"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":162738,"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-4d30609870afaf16"]}],"region_exact":{},"incoming_edges":[{"id":"e-4d30609870afaf16","operation":{"operation":"obreschkoff_combination"},"operation_detail":null,"conclusion_id":"n-a712a218f238860b","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":"671cd8aa60d26fbbcdaeb99a019615a46682575499bf06c95c26210a851f2b3b","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{}}],"premises":[{"position":0,"node_id":"n-a98fae1bf9eeb913","role":"sequence","name":"Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by the n-th Catalan number (A000108).","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-a98fae1bf9eeb913","role":"sequence","name":"Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by the n-th Catalan number (A000108).","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":[],"oeis":[],"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:     2,      1;\n   2:     3,      5,      2;\n   3:     7,     19,     17,       5;\n   4:    19,     71,     99,      61,      14;\n   5:    56,    266,    504,     476,     224,      42;\n   6:   174,   1002,   2400,    3060,    2190,     834,     132;\n   7:   561,   3795,  10989,   17655,   16995,    9801,    3135,     429;\n   8:  1859,  14443,  49049,   95095,  115115,   89089,   43043,   11869,   1430;\n   9:  6292,  55198, 215072,  488488,  712712,  692692,  448448,  186472,  45188,   4862;\n  10: 21658, 211718, 930852, 2423928, 4139772, 4845204, 3935568, 2190552, 799578, 172822, 16796;\n","recurrence_sentence":"The row polynomials satisfy (5*n^2-n-6+(4*n^2-2*n-6)*x)*P(n,x) = (20*n^2-34*n+6+(36*n^2-66*n+18)*x+(16*n^2-32*n+12)*x^2)*P(n-1,x) for n >= 2, with P(1,x) = 2 + x.","mathematica":"P[1, x_] := 2 + x;\nP[n_, x_] := P[n, x] = Expand[Together[((20*n^2 - 34*n + 6 + (36*n^2 - 66*n + 18)*x + (16*n^2 - 32*n + 12)*x^2)*P[n - 1, x])/(5*n^2 - n - 6 + (4*n^2 - 2*n - 6)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 2 + x; for(n = 2, N, my(i = n); v[i] = ((20*n^2 - 34*n + 6 + (36*n^2 - 66*n + 18)*x + (16*n^2 - 32*n + 12)*x^2)*v[i - 1])/(5*n^2 - n - 6 + (4*n^2 - 2*n - 6)*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-a98fae1bf9eeb913","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-4d30609870afaf16"],"text":"A085880 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A085880 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}