{"id":"n-df199653ca102fb5","revision":0,"first_index":0,"fingerprint":"f6-5344ee972d4f418a","name":"shift_x_plus_one of reversal of Triangular array read by rows. T(n,k) = A008277(n,k)*2^k; n >= 1, 1 <= k <= n.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["2"],["6","2"],["22","16","2"],["94","110","34","2"],["454","748","392","68","2"],["2430","5226","3964","1236","134","2"],["14214","38008","38278","17792","3698","264","2"],["89918","288998","366062","229974","73546","10818","522","2"],["610182","2299204","3529436","2822084","1242872","291052","31396","1036","2"],["4412798","19125602","34633432","33844984","19205860","6317036","1125560","91064","2062","2"],["33827974","166117136","347662922","403313920","282153148","121115872","30947732","4303616","264926","4112","2"],["273646526","1504070366","3580195898","4823551146","4031697740","2152733260","727517652","148315636","16382678","774198","8210","2"],["2326980998","14172210908","37876079568","58263577188","56838480454","36499409048","15485958672","4234044488","701845042","62351820","2273312","16404","2"],["20732504062","138737335642","411912674580","713655148124","798008089818","601127617294","308814836984","107020994856","24141255506","3299436630","237860276","6705084","32790","2"],["192982729350","1408752406696","4605572013934","8887471216928","11228633261438","9733957580040","5896260939590","2491430641536","719452034322","135870057016","15471986122","910817120","19852906","65560","2"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["2","8","40","240","1664","12992","112256","1059840","10827264","118758400","1389711360","17258893312","226463227904","3127694491648","45316785602560"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["2"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["2","0"],["1","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-1","-2","-1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":178822,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, -1]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, -1]","region_exact":null,"edges":["e-ed9047c4c3c14788"]},{"id":178823,"position":1,"claim":"one_signed_coefficients","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":0},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 0: the coefficients of P_n have one sign","region_exact":null,"edges":["e-ed9047c4c3c14788"]},{"id":178824,"position":2,"claim":"hadamard_kernel","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":0},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-ed9047c4c3c14788"]},{"id":178825,"position":3,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1","region_exact":null,"edges":["e-ed9047c4c3c14788"]}],"region_exact":{},"incoming_edges":[{"id":"e-ed9047c4c3c14788","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-df199653ca102fb5","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":"98afa20ff034f910d0f855ffb91d04bd4052813d8f999ae50069d9c09746c59a","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-f03980d46a617c93","role":"sequence","name":"reversal of Triangular array read by rows. T(n,k) = A008277(n,k)*2^k; n >= 1, 1 <= k <= n.","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-6d53fbe74c82d499","kind":"fingerprint_related","node_a_id":"n-2781c74a5cedd697","node_b_id":"n-df199653ca102fb5","fingerprint":"f6-5344ee972d4f418a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-5344ee972d4f418a"},"other_id":"n-2781c74a5cedd697","other":{"id":"n-2781c74a5cedd697","name":"mobius_x_over_x_plus_one of Triangular array read by rows. T(n,k) = A008277(n,k)*2^k; n >= 1, 1 <= k <= n.","first_index":0,"fingerprint":"f6-5344ee972d4f418a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["2"],["2","6"],["2","16","22"]],"card_rows":[["2"],["2","6"],["2","16","22"],["2","34","110","94"]],"row_sums_preview":["2","8","40","240","1664","12992","112256","1059840"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-d5fb64ec7bd8ef70","steps":["x \u21a6 x/(x+1)"],"edges":["e-12a33f06274c1738"],"text":"A227450 \u2192 x \u21a6 x/(x+1)"},"headline":"A227450 \u2192 x \u21a6 x/(x+1)","chain":null}}],"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:       2;\n  1:       6,        2;\n  2:      22,       16,        2;\n  3:      94,      110,       34,        2;\n  4:     454,      748,      392,       68,        2;\n  5:    2430,     5226,     3964,     1236,      134,       2;\n  6:   14214,    38008,    38278,    17792,     3698,     264,       2;\n  7:   89918,   288998,   366062,   229974,    73546,   10818,     522,     2;\n  8:  610182,  2299204,  3529436,  2822084,  1242872,  291052,   31396,  1036,    2;\n  9: 4412798, 19125602, 34633432, 33844984, 19205860, 6317036, 1125560, 91064, 2062, 2;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (n+2+n*x)*P(n-1,x) - (1+2*x+x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 2.","mathematica":"P[0, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[(n + 2 + n*x)*P[n - 1, x] - (1 + 2*x + x^2)*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] = 2; for(n = 1, N, my(i = n + 1); v[i] = (n + 2 + n*x)*v[i - 1] - (1 + 2*x + x^2)*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-d5fb64ec7bd8ef70","steps":["reverse","x \u21a6 x+1"],"edges":["e-88af8bd6d056b310","e-ed9047c4c3c14788"],"text":"A227450 \u2192 reverse \u2192 x \u21a6 x+1"},"headline":"A227450 \u2192 reverse \u2192 x \u21a6 x+1","chain":null}
