{"id":"n-98104e547dded209","revision":1,"first_index":0,"fingerprint":"f6-ca870e8eb05a74d5","name":"obreschkoff_combination of Triangle of generalized Stirling numbers of the first kind read by rows: T(n, k) = (-1)^(n+k)*Sum_{m=0..n} Stirling1(n, m) * binomial(m, k), where Stirling1 is A048994.","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":true,"rows":[["0","1"],["-1","0","1"],["0","-2","1","1"],["0","-3","-1","3","1"],["0","-8","-6","7","6","1"],["0","-30","-31","20","30","10","1"],["0","-144","-180","64","165","79","15","1"],["0","-840","-1198","189","1029","630","168","21","1"],["0","-5760","-9072","76","7252","5369","1792","314","28","1"],["0","-45360","-77292","-8604","57203","49644","19551","4284","537","36","1"],["0","-403200","-732960","-154664","499860","499190","223965","57813","9090","860","45","1"],["0","-3991680","-7663536","-2271060","4793096","5446925","2720927","798105","148203","17655","1309","55","1"],["0","-43545600","-87626880","-32499072","50003976","64254916","35171070","11446259","2419758","341583","32010","1913","66","1"],["0","-518918400","-1088068320","-475490808","563204564","816070398","483792881","171781896","40343732","6501924","724438","54912","2704","78","1"],["0","-6706022400","-14583179520","-7239074688","6800508624","11112857392","7072721656","2706255552","693970277","124543991","15888873","1436435","89999","3717","91","1"]],"degrees":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],"row_sums":["1","0","0","0","0","0","0","0","0","0","0","0","0","0","0"],"nonn":false,"definitions":[{"position":0,"source":"derived:obreschkoff_combination","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0","1"]],"recurrence":{"denominator":{"coeffs":[["-1","1"],["1","0"]]},"terms":[{"coeff":{"coeffs":[["0","-2","1"],["-2","2","0"],["1","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":162709,"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-c8fbb9d7e751998c"]}],"region_exact":{},"incoming_edges":[{"id":"e-c8fbb9d7e751998c","operation":{"operation":"obreschkoff_combination"},"operation_detail":null,"conclusion_id":"n-98104e547dded209","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":"2f54a4cc4544c8cf53c194367bd12e3fb22eff937b807222ba1584942170351e","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-af47a5bf0b42b546","role":"sequence","name":"Triangle of generalized Stirling numbers of the first kind read by rows: T(n, k) = (-1)^(n+k)*Sum_{m=0..n} Stirling1(n, m) * binomial(m, k), where Stirling1 is A048994.","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-af47a5bf0b42b546","role":"sequence","name":"Triangle of generalized Stirling numbers of the first kind read by rows: T(n, k) = (-1)^(n+k)*Sum_{m=0..n} Stirling1(n, m) * binomial(m, k), where Stirling1 is A048994.","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:  0,      1;\n  1: -1,      0,      1;\n  2:  0,     -2,      1,     1;\n  3:  0,     -3,     -1,     3,     1;\n  4:  0,     -8,     -6,     7,     6,     1;\n  5:  0,    -30,    -31,    20,    30,    10,     1;\n  6:  0,   -144,   -180,    64,   165,    79,    15,    1;\n  7:  0,   -840,  -1198,   189,  1029,   630,   168,   21,   1;\n  8:  0,  -5760,  -9072,    76,  7252,  5369,  1792,  314,  28,  1;\n  9:  0, -45360, -77292, -8604, 57203, 49644, 19551, 4284, 537, 36, 1;\n","recurrence_sentence":"The row polynomials satisfy (n-1+x)*P(n,x) = (n^2-2*n+(2*n-2)*x+x^2)*P(n-1,x) for n >= 1, with P(0,x) = x.","mathematica":"P[0, x_] := x;\nP[n_, x_] := P[n, x] = Expand[Together[((n^2 - 2*n + (2*n - 2)*x + x^2)*P[n - 1, x])/(n - 1 + x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = x; for(n = 1, N, my(i = n + 1); v[i] = ((n^2 - 2*n + (2*n - 2)*x + x^2)*v[i - 1])/(n - 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-4f6e747b794bcc2e","steps":["x \u21a6 2x\u22123","P\u2099 + P\u2099\u208a\u2081"],"edges":["e-0658035b499d17ce","e-c8fbb9d7e751998c"],"text":"A028338 \u2192 x \u21a6 2x\u22123 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A028338 \u2192 x \u21a6 2x\u22123 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}