{"id":"n-eea87339949eb904","revision":0,"first_index":1,"fingerprint":"f6-5e6f94e8e9fb2fe6","name":"shift_x_plus_one of Triangle, T(n, k) = (1/2)*(n+2)! * [x^k]( p(x, n) ), where p(x,0) = 1, p(x,1) = -x, P(x, n) = (1/(n+1))*( (2*n-x)*P(x, n-1) - n*P(x, n-2) ), read by rows.","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":[["-20","-8","4"],["-80","20","35","-5"],["-192","456","174","-84","6"],["784","4312","70","-910","161","-7"],["19072","31424","-12400","-7488","2776","-272","8"],["229536","166464","-221112","-38232","38880","-6660","423","-9"],["2299840","5440","-2878160","207520","468400","-132440","13810","-620","10"],["20282944","-18908032","-31937488","10831568","4681688","-2332528","362054","-25894","869","-11"],["144429312","-414691584","-295970880","234447360","29014368","-37561440","8391768","-857280","45060","-1176","12"],["385305856","-6886850048","-1717027520","4105398336","-263389152","-552403488","179830560","-24891360","1829568","-73996","1547","-13"],["-14720394752","-101328736768","14610264704","64149508864","-15819558720","-7088637696","3639343680","-666180480","64552992","-3605560","115990","-1988","14"],["-469430929920","-1355454981120","834733246080","902083885440","-441108912000","-65206016640","70010429760","-16889601600","2086097040","-151312200","6668310","-174990","2505","-15"],["-10222242357248","-15838740094976","22213436839936","10720143785984","-10122578323456","180437737472","1270640477184","-411873890304","63635299584","-5775316096","327465248","-11707264","255664","-3104","16"],["-195267955503104","-131529026486272","488423206231040","76681262976000","-210809324804096","32942511497728","21107235018368","-9734071344000","1866423931680","-206979822016","14525877256","-664039448","19677840","-363460","3791","-17"],["-3465377681743872","451064583806976","9773569693255680","-1087605702807552","-4085395441330176","1252533299973120","291989185705728","-223235733726720","53242318312128","-7106730814656","601635673584","-33804766944","1275365520","-31871448","504666","-4572","18"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["-24","-30","360","4410","33120","169290","-16200","-17044830","-332754480","-4772269710","-57163329720","-539321599350","-2152599831360","73487162851290","2954246865531480"],"nonn":false,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["-20","-8","4"],["-80","20","35","-5"]],"recurrence":{"denominator":{"coeffs":[["2"],["1"]]},"terms":[{"coeff":{"coeffs":[["-6"],["-11"],["-6"],["-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["3","-3"],["7","-1"],["2","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":188364,"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-5b14554a44082fcc"]},{"id":188365,"position":1,"claim":"interlacing","from_index":2,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2","region_exact":null,"edges":["e-5b14554a44082fcc"]}],"region_exact":{},"incoming_edges":[{"id":"e-5b14554a44082fcc","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-eea87339949eb904","premise_count":1,"theorem":"a real affine substitution x -> a x + b (a != 0), the reversal x^d p(1/x) (d >= deg p) and a real Mobius map (c x + e)^d p((a x + b)/(c x + e)) (a e - b c != 0, d >= deg p) map real roots to real roots: a root r of p gives the root (r - b)/a, 1/r (and 0) or (e r - b)/(a - c r) (and -e/c); for an input row that is a polynomial the degree condition holds automatically","lean_route":null,"lean_declarations":["Polynomial.Splits.comp_of_degree_le_one"],"certificate_hash":"eaeba238b5516d27b24d15f3292f821a351c446a6f90eb349bfc9e317dfef9cc","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-cfb1b41332b76420","role":"sequence","name":"Triangle, T(n, k) = (1/2)*(n+2)! * [x^k]( p(x, n) ), where p(x,0) = 1, p(x,1) = -x, P(x, n) = (1/(n+1))*( (2*n-x)*P(x, n-1) - n*P(x, n-2) ), read by rows.","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)"}]}],"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:       -20,          -8,           4;\n   2:       -80,          20,          35,         -5;\n   3:      -192,         456,         174,        -84,          6;\n   4:       784,        4312,          70,       -910,        161,         -7;\n   5:     19072,       31424,      -12400,      -7488,       2776,       -272,         8;\n   6:    229536,      166464,     -221112,     -38232,      38880,      -6660,       423,        -9;\n   7:   2299840,        5440,    -2878160,     207520,     468400,    -132440,     13810,      -620,      10;\n   8:  20282944,   -18908032,   -31937488,   10831568,    4681688,   -2332528,    362054,    -25894,     869,    -11;\n   9: 144429312,  -414691584,  -295970880,  234447360,   29014368,  -37561440,   8391768,   -857280,   45060,  -1176,   12;\n  10: 385305856, -6886850048, -1717027520, 4105398336, -263389152, -552403488, 179830560, -24891360, 1829568, -73996, 1547, -13;\n","recurrence_sentence":"The row polynomials satisfy (n+2)*P(n,x) = -(n^3+6*n^2+11*n+6)*P(n-2,x) + (2*n^2+7*n+3-(n+3)*x)*P(n-1,x) for n >= 3, with P(1,x) = -20 - 8*x + 4*x^2, P(2,x) = -80 + 20*x + 35*x^2 - 5*x^3.","mathematica":"P[1, x_] := -20 - 8*x + 4*x^2;\nP[2, x_] := -80 + 20*x + 35*x^2 - 5*x^3;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^3 + 6*n^2 + 11*n + 6)*P[n - 2, x] + (2*n^2 + 7*n + 3 - (n + 3)*x)*P[n - 1, x])/(n + 2)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = -20 - 8*x + 4*x^2; v[2] = -80 + 20*x + 35*x^2 - 5*x^3; for(n = 3, N, my(i = n); v[i] = (-(n^3 + 6*n^2 + 11*n + 6)*v[i - 2] + (2*n^2 + 7*n + 3 - (n + 3)*x)*v[i - 1])/(n + 2)); 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-cfb1b41332b76420","steps":["x \u21a6 x+1"],"edges":["e-5b14554a44082fcc"],"text":"A136532 \u2192 x \u21a6 x+1"},"headline":"A136532 \u2192 x \u21a6 x+1","chain":null}