{"id":"n-4368f011258c3395","revision":1,"first_index":0,"fingerprint":"f6-de2a5e16d4eb9421","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) * 2^(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":[["-1","1"],["0","-2","1"],["2","-1","-2","1"],["0","4","-4","-1","1"],["0","6","-1","-7","1","1"],["0","16","4","-20","-5","4","1"],["0","60","32","-71","-40","10","8","1"],["0","288","216","-308","-266","7","49","13","1"],["0","1680","1556","-1576","-1869","-231","294","126","19","1"],["0","11520","12384","-9224","-14428","-3486","1785","1164","258","26","1"],["0","90720","109224","-60084","-123010","-42085","10542","10983","3210","465","34","1"],["0","806400","1062720","-423632","-1154384","-498520","51260","108339","39633","7370","770","43","1"],["0","7983360","11335392","-3121416","-11857252","-6100754","5071","1124717","501699","112893","15037","1199","53","1"],["0","87091200","131708160","-22628736","-132507024","-78505856","-6087224","12278552","6606743","1736592","277563","28184","1781","64","1"],["0","1037836800","1657218240","-137086704","-1601899936","-1068936232","-151515364","140229089","91094432","27339884","5053048","614614","49504","2548","76","1"]],"degrees":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],"row_sums":["0","-1","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":[["-1","1"]],"recurrence":{"denominator":{"coeffs":[["-2","1"],["1","0"]]},"terms":[{"coeff":{"coeffs":[["3","-4","1"],["-4","2","0"],["1","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":162704,"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-1e544bff83909e45"]}],"region_exact":{},"incoming_edges":[{"id":"e-1e544bff83909e45","operation":{"operation":"obreschkoff_combination"},"operation_detail":null,"conclusion_id":"n-4368f011258c3395","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":"ae5a62a420c5a300fe64fa0c0ae5c00d919dc26ab3d3245b6f46de1400e7f985","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-5baee50acd24e741","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) * 2^(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-5baee50acd24e741","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) * 2^(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: -1,     1;\n  1:  0,    -2,     1;\n  2:  2,    -1,    -2,     1;\n  3:  0,     4,    -4,    -1,      1;\n  4:  0,     6,    -1,    -7,      1,     1;\n  5:  0,    16,     4,   -20,     -5,     4,    1;\n  6:  0,    60,    32,   -71,    -40,    10,    8,    1;\n  7:  0,   288,   216,  -308,   -266,     7,   49,   13,   1;\n  8:  0,  1680,  1556, -1576,  -1869,  -231,  294,  126,  19,  1;\n  9:  0, 11520, 12384, -9224, -14428, -3486, 1785, 1164, 258, 26, 1;\n","recurrence_sentence":"The row polynomials satisfy (n-2+x)*P(n,x) = (n^2-4*n+3+(2*n-4)*x+x^2)*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[Together[((n^2 - 4*n + 3 + (2*n - 4)*x + x^2)*P[n - 1, x])/(n - 2 + 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] = ((n^2 - 4*n + 3 + (2*n - 4)*x + x^2)*v[i - 1])/(n - 2 + 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\u22125","P\u2099 + P\u2099\u208a\u2081"],"edges":["e-2198daa4fdd0bfd0","e-1e544bff83909e45"],"text":"A028338 \u2192 x \u21a6 2x\u22125 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A028338 \u2192 x \u21a6 2x\u22125 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}