{"id":"n-24c1a22c5b1801ba","revision":1,"first_index":0,"fingerprint":"f6-4c22959b3ea620d2","name":"obreschkoff_combination of Triangle read by rows: T(n,k) = Sum_{i=0..n-1} binomial(n-1, i)*T(n-1-i,k-1) - Sum_{i=1..n-1} binomial(n-1,i)*T(n-1-i,k) for 1 <= k <= n+1 with T(0,1) = 1 (and T(n,k) = 0 otherwise).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":1,"palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":false,"rows":[["1","1"],["-1","2","1"],["-2","-1","4","1"],["1","-11","4","7","1"],["11","-18","-24","21","11","1"],["18","41","-129","-5","60","16","1"],["-41","317","-202","-384","155","134","22","1"],["-317","680","1065","-1974","-434","693","259","29","1"],["-680","-1767","8732","-3121","-7175","1477","2044","454","37","1"],["1767","-19911","25060","28069","-39206","-12054","10563","4926","741","46","1"],["19911","-68264","-53998","266091","-68485","-162854","16842","39117","10440","1145","56","1"],["68264","59643","-974533","1018215","806421","-983807","-335621","207141","112332","20185","1694","67","1"],["-59643","2076973","-4944068","-1145572","9162934","-2098888","-4447520","215710","924132","277057","36388","2419","79","1"],["-2076973","11905466","-4374447","-45032520","46000604","25353614","-30293978","-10330606","5115253","3053765","614328","62049","3354","92","1"],["-11905466","18577387","138254132","-323474423","12201826","354532542","-84096012","-143530244","3107533","26455858","8514506","1256619","101101","4536","106","1"]],"degrees":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],"row_sums":["2","2","2","2","2","2","2","2","2","2","2","2","2","2","2"],"nonn":false,"definitions":[{"position":0,"source":"derived:obreschkoff_combination","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-1","1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":162706,"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-cb47efc3e6c3b2c6"]}],"region_exact":{},"incoming_edges":[{"id":"e-cb47efc3e6c3b2c6","operation":{"operation":"obreschkoff_combination"},"operation_detail":null,"conclusion_id":"n-24c1a22c5b1801ba","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":"5760bfa837affd15f974a8053fe9c46339bb7603bde2d64c454e092e22ee3d3b","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-13114c927e22057d","role":"sequence","name":"Triangle read by rows: T(n,k) = Sum_{i=0..n-1} binomial(n-1, i)*T(n-1-i,k-1) - Sum_{i=1..n-1} binomial(n-1,i)*T(n-1-i,k) for 1 <= k <= n+1 with T(0,1) = 1 (and T(n,k) = 0 otherwise).","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-13114c927e22057d","role":"sequence","name":"Triangle read by rows: T(n,k) = Sum_{i=0..n-1} binomial(n-1, i)*T(n-1-i,k-1) - Sum_{i=1..n-1} binomial(n-1,i)*T(n-1-i,k) for 1 <= k <= n+1 with T(0,1) = 1 (and T(n,k) = 0 otherwise).","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":[],"oeis":[],"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:    1,      1;\n  1:   -1,      2,     1;\n  2:   -2,     -1,     4,     1;\n  3:    1,    -11,     4,     7,      1;\n  4:   11,    -18,   -24,    21,     11,      1;\n  5:   18,     41,  -129,    -5,     60,     16,     1;\n  6:  -41,    317,  -202,  -384,    155,    134,    22,    1;\n  7: -317,    680,  1065, -1974,   -434,    693,   259,   29,   1;\n  8: -680,  -1767,  8732, -3121,  -7175,   1477,  2044,  454,  37,  1;\n  9: 1767, -19911, 25060, 28069, -39206, -12054, 10563, 4926, 741, 46, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = x*P(n-1,x) - (1-x)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1 + x.","mathematica":"P[0, x_] := 1 + x;\nP[n_, x_] := P[n, x] = Expand[x*P[n - 1, x] - (1 - x)*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1 + x; for(n = 1, N, my(i = n + 1); v[i] = x*v[i - 1] - (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 - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-6fea5aff4f44fa54","steps":["basis A341287","P\u2099 + P\u2099\u208a\u2081"],"edges":["e-0f172f031b36afc5","e-cb47efc3e6c3b2c6"],"text":"A023531 \u2192 basis A341287 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A023531 \u2192 basis A341287 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}