{"id":"n-ed9cc69a9e3761ed","revision":1,"first_index":1,"fingerprint":"f6-f40e790c2f983509","name":"obreschkoff_combination of Triangle of Stirling numbers of first kind, s(n, n-k+1), n >= 1, 1 <= k <= n. Also triangle T(n,k) giving coefficients in expansion of n!*binomial(x,n)/x in powers of x.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":true,"rows":[["2","-1"],["2","-4","2"],["2","-9","13","-6"],["2","-16","46","-56","24"],["2","-25","120","-275","298","-120"],["2","-36","260","-960","1898","-1884","720"],["2","-49","497","-2695","8393","-14896","13788","-5040"],["2","-64","868","-6496","29218","-80416","131192","-114624","40320"],["2","-81","1416","-13986","85722","-336609","841804","-1282284","1066896","-362880"],["2","-100","2190","-27600","221046","-1171380","4140610","-9582200","13780152","-10991520","3628800"],["2","-121","3245","-50820","515196","-3539613","16756465","-54405230","118011652","-161546616","124172640","-39916800"],["2","-144","4642","-88440","1106886","-9564192","58329766","-252065880","762464912","-1564932864","2052103392","-1526359680","479001600"],["2","-169","6448","-146861","2223936","-23596287","180026704","-997013303","3993325622","-11371718644","22245312648","-28083159936","20281760640","-6227020800"],["2","-196","8736","-234416","4224220","-53981928","503448088","-3472396928","17745441714","-66621070516","180035358776","-337585977456","411959556864","-289692668160","87178291200"],["2","-225","11585","-361725","7649369","-115870755","1296507355","-10889081775","69040452481","-329466577440","1169393712760","-3017802606000","5449152501648","-6449283262080","4426341292800","-1307674368000"]],"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":1,"initial_polynomials":[["2","-1"]],"recurrence":{"denominator":{"coeffs":[["-2","-1"],["0","1"]]},"terms":[{"coeff":{"coeffs":[["-2","-2","0"],["0","3","1"],["0","0","-1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":162737,"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-ed573324c4b51ce2"]}],"region_exact":{},"incoming_edges":[{"id":"e-ed573324c4b51ce2","operation":{"operation":"obreschkoff_combination"},"operation_detail":null,"conclusion_id":"n-ed9cc69a9e3761ed","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":"799fec5a3367e3f659555e115bd610eec1a89fe03e62a55ad496494c595bcc42","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{}}],"premises":[{"position":0,"node_id":"n-a6c3537128a476c6","role":"sequence","name":"Triangle of Stirling numbers of first kind, s(n, n-k+1), n >= 1, 1 <= k <= n. Also triangle T(n,k) giving coefficients in expansion of n!*binomial(x,n)/x in powers of x.","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-a6c3537128a476c6","role":"sequence","name":"Triangle of Stirling numbers of first kind, s(n, n-k+1), n >= 1, 1 <= k <= n. Also triangle T(n,k) giving coefficients in expansion of n!*binomial(x,n)/x in powers of x.","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: 2,   -4,    2;\n   3: 2,   -9,   13,     -6;\n   4: 2,  -16,   46,    -56,     24;\n   5: 2,  -25,  120,   -275,    298,     -120;\n   6: 2,  -36,  260,   -960,   1898,    -1884,     720;\n   7: 2,  -49,  497,  -2695,   8393,   -14896,   13788,    -5040;\n   8: 2,  -64,  868,  -6496,  29218,   -80416,  131192,  -114624,    40320;\n   9: 2,  -81, 1416, -13986,  85722,  -336609,  841804, -1282284,  1066896,   -362880;\n  10: 2, -100, 2190, -27600, 221046, -1171380, 4140610, -9582200, 13780152, -10991520, 3628800;\n","recurrence_sentence":"The row polynomials satisfy -(2-(n-1)*x)*P(n,x) = -(2-(3*n-2)*x+(n^2-n)*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[(-(2 - (3*n - 2)*x + (n^2 - n)*x^2)*P[n - 1, x])/(-2 + (n - 1)*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] = (-(2 - (3*n - 2)*x + (n^2 - n)*x^2)*v[i - 1])/(-2 + (n - 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-7fd16a32c438d522","steps":["x \u21a6 \u2212x\u22121 (shift -1)","P\u2099 + P\u2099\u208a\u2081"],"edges":["e-68dd2f09e1ded893","e-ed573324c4b51ce2"],"text":"A059364 \u2192 x \u21a6 \u2212x\u22121 (shift -1) \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A059364 \u2192 x \u21a6 \u2212x\u22121 (shift -1) \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}