{"id":"n-ee5b69bc1fbff350","revision":2,"first_index":0,"fingerprint":"f6-397861419350734b","name":"basis_map_A008288 of Triangular array read by rows. T(n, k) is the coefficient of x^k in a(n+3) where a(1) = a(2) = a(3) = 1 and a(m+2) = (m*x + 2)*a(m+1) - a(m) for all m in Z.","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":[["1"],["2","1"],["7","10","2"],["33","88","46","6"],["191","778","762","244","24"],["1304","7263","11268","6492","1508","120"],["10241","72968","161112","144476","58600","10728","720"],["90865","792816","2317324","2952204","1827472","571648","86616","5040"],["898409","9309024","34244032","58117724","50577688","23754776","6050912","783648","40320"],["9791634","117777801","525759196","1133657668","1309445980","855140280","322796224","69400368","7859232","362880"],["116601199","1599857258","8434591330","22292840772","32734259032","28189449744","14650581776","4618574848","859608864","86575680","3628800"],["1506023953","23245592920","141755563834","446732987522","806492927952","880239002736","599141849392","257857225792","69761912672","11453317344","1039423680","39916800"],["20967734143","359983018482","2497892961010","9184641805096","19853722782944","26625375145104","22870364993808","12808426725568","4696657683616","1112770468608","163503191808","13510920960","479001600"],["313009988192","5921660772015","46141036248888","194544035500456","492936734086328","792309016602832","834310602122016","586968251257936","278463725915616","88868444080960","18729144218688","2491353321984","189054017280","6227020800"],["4987192076929","103150869003120","892800908974640","4255955277966456","12423430454347744","23451681988650784","29573521844161232","25429703318818784","15088225309367808","6197907764587968","1750073762531840","332164665063552","40375731442176","2833566865920","87178291200"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","3","19","173","1999","27955","458845","8643985","183776533","4351991263","113586569303","3239265744597","100208296432747","3342192295153991","119546899832150153"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A008288","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["2","1"],["7","10","2"],["33","88","46","6"]],"recurrence":{"denominator":{"coeffs":[["2"],["-3"],["1"]]},"terms":[{"coeff":{"coeffs":[["0"],["1"],["-1"]]},"deriv_order":0,"offset":4,"sign":"none"},{"coeff":{"coeffs":[["0","0"],["-4","2"],["1","-3"],["1","1"]]},"deriv_order":0,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["-2","0"],["8","-6"],["0","11"],["-2","-6"],["0","1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["4","0"],["-8","2"],["1","-3"],["1","1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":166223,"position":0,"claim":"real_rooted","from_index":0,"region_key":"[-1564556493/268435456, -46056243/268435456]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"-1564556493/268435456"},"upper":{"closed":true,"value":"-46056243/268435456"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in [-1564556493/268435456, -46056243/268435456]","region_exact":null,"edges":["e-393d995dcff5ebfb"]},{"id":166224,"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-393d995dcff5ebfb","e-fca58ee161653196"]},{"id":166225,"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-393d995dcff5ebfb","e-fca58ee161653196"]},{"id":168716,"position":3,"claim":"real_rooted","from_index":0,"region_key":"[-1564556493/268435456, -46056243/268435456]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"exact":{"radicand":"2","rational_part":"-3","sqrt_coefficient":"-2"},"value":"-1564556493/268435456"},"upper":{"closed":true,"exact":{"radicand":"2","rational_part":"-3","sqrt_coefficient":"2"},"value":"-46056243/268435456"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in [\u22123\u22122\u221a2, \u22123+2\u221a2] (rational outer bound [-1564556493/268435456, -46056243/268435456])","region_exact":"[\u22123\u22122\u221a2, \u22123+2\u221a2]","edges":["e-fca58ee161653196"]}],"region_exact":{"[-1564556493/268435456, -46056243/268435456]":"[\u22123\u22122\u221a2, \u22123+2\u221a2]"},"incoming_edges":[{"id":"e-393d995dcff5ebfb","operation":{"map":"A008288","operation":"basis_map"},"operation_detail":"A008288","conclusion_id":"n-ee5b69bc1fbff350","premise_count":1,"theorem":"affine current multiplier with an affine initial row (module documentation, \"The affine-current theorem\"): with beta = b/d, K = beta^2 + a beta - c > 0 and eta = m/d >= 0, every real polynomial with all roots in E = [-beta - sqrt(K), -beta + sqrt(K)] has a real-rooted image, and no interval of positive length outside E works (Theorem 1); for eta > 0 the images lie in I = [(-a - 2 beta - 2 sqrt(K))/d, (-a - 2 beta + 2 sqrt(K))/d] if and only if eta >= 1/2 and (s - (2 eta - 1) beta)^2 <= (2 eta - 1)^2 K, s = l - eta a (Theorem 2); with a >= 0, K >= beta^2, a + 2 beta >= 0, a^2 + 4c >= 0, l >= 0 and the scalar criterion, PF inputs with roots in [-R, 0], R = beta + sqrt(K), have PF images, sharply (the common-interlacer proof; the Chebyshev case is Iserles and Saff, Math. Proc. Cambridge Philos. Soc. 105 (1989), Theorem 1; the scalar criterion is RealRooted orderTwoConstants_nonneg_iff)","lean_route":null,"lean_declarations":[],"certificate_hash":"9316c9bc3c62cfff21e9b43a5af76f4558896ee87fd65a1ce4835f0d0021fc72","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"-1564556493/268435456"},"upper":{"closed":true,"value":"-46056243/268435456"}}}],"premises":[{"position":0,"node_id":"n-f3e0631eb8740f76","role":"sequence","name":"Triangular array read by rows. T(n, k) is the coefficient of x^k in a(n+3) where a(1) = a(2) = a(3) = 1 and a(m+2) = (m*x + 2)*a(m+1) - a(m) for all m in Z.","uses":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"-2"},"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in [-2, 0]"}]},{"id":"e-fca58ee161653196","operation":{"map":"A008288","operation":"basis_map"},"operation_detail":"A008288","conclusion_id":"n-ee5b69bc1fbff350","premise_count":1,"theorem":"affine current multiplier with an affine initial row (module documentation, \"The affine-current theorem\"): with beta = b/d, K = beta^2 + a beta - c > 0 and eta = m/d >= 0, every real polynomial with all roots in E = [-beta - sqrt(K), -beta + sqrt(K)] has a real-rooted image, and no interval of positive length outside E works (Theorem 1); for eta > 0 the images lie in I = [(-a - 2 beta - 2 sqrt(K))/d, (-a - 2 beta + 2 sqrt(K))/d] if and only if eta >= 1/2 and (s - (2 eta - 1) beta)^2 <= (2 eta - 1)^2 K, s = l - eta a (Theorem 2); with a >= 0, K >= beta^2, a + 2 beta >= 0, a^2 + 4c >= 0, l >= 0 and the scalar criterion, PF inputs with roots in [-R, 0], R = beta + sqrt(K), have PF images, sharply (the common-interlacer proof; the Chebyshev case is Iserles and Saff, Math. Proc. Cambridge Philos. Soc. 105 (1989), Theorem 1; the scalar criterion is RealRooted orderTwoConstants_nonneg_iff)","lean_route":null,"lean_declarations":[],"certificate_hash":"ab071cd420dc37d5df449d8c8c3041c916d8ab89392fb98a15abe19665f7c874","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"exact":{"radicand":"2","rational_part":"-3","sqrt_coefficient":"-2"},"value":"-1564556493/268435456"},"upper":{"closed":true,"exact":{"radicand":"2","rational_part":"-3","sqrt_coefficient":"2"},"value":"-46056243/268435456"}}}],"premises":[{"position":0,"node_id":"n-f3e0631eb8740f76","role":"sequence","name":"Triangular array read by rows. T(n, k) is the coefficient of x^k in a(n+3) where a(1) = a(2) = a(3) = 1 and a(m+2) = (m*x + 2)*a(m+1) - a(m) for all m in Z.","uses":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"-2"},"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in [-2, 0]"}]}],"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;\n  1:       2,         1;\n  2:       7,        10,         2;\n  3:      33,        88,        46,          6;\n  4:     191,       778,       762,        244,         24;\n  5:    1304,      7263,     11268,       6492,       1508,       120;\n  6:   10241,     72968,    161112,     144476,      58600,     10728,       720;\n  7:   90865,    792816,   2317324,    2952204,    1827472,    571648,     86616,     5040;\n  8:  898409,   9309024,  34244032,   58117724,   50577688,  23754776,   6050912,   783648,   40320;\n  9: 9791634, 117777801, 525759196, 1133657668, 1309445980, 855140280, 322796224, 69400368, 7859232, 362880;\n","recurrence_sentence":"The row polynomials satisfy (n^2-3*n+2)*P(n,x) = -(n^2-n)*P(n-4,x) + (n^3+n^2-4*n+(n^3-3*n^2+2*n)*x)*P(n-3,x) - (2*n^3-8*n+2-(n^4-6*n^3+11*n^2-6*n)*x)*P(n-2,x) + (n^3+n^2-8*n+4+(n^3-3*n^2+2*n)*x)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 2 + x, P(2,x) = 7 + 10*x + 2*x^2, P(3,x) = 33 + 88*x + 46*x^2 + 6*x^3.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + x;\nP[2, x_] := 7 + 10*x + 2*x^2;\nP[3, x_] := 33 + 88*x + 46*x^2 + 6*x^3;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^2 - n)*P[n - 4, x] + (n^3 + n^2 - 4*n + (n^3 - 3*n^2 + 2*n)*x)*P[n - 3, x] - (2*n^3 - 8*n + 2 - (n^4 - 6*n^3 + 11*n^2 - 6*n)*x)*P[n - 2, x] + (n^3 + n^2 - 8*n + 4 + (n^3 - 3*n^2 + 2*n)*x)*P[n - 1, x])/(n^2 - 3*n + 2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + x; v[3] = 7 + 10*x + 2*x^2; v[4] = 33 + 88*x + 46*x^2 + 6*x^3; for(n = 4, N, my(i = n + 1); v[i] = (-(n^2 - n)*v[i - 4] + (n^3 + n^2 - 4*n + (n^3 - 3*n^2 + 2*n)*x)*v[i - 3] - (2*n^3 - 8*n + 2 - (n^4 - 6*n^3 + 11*n^2 - 6*n)*x)*v[i - 2] + (n^3 + n^2 - 8*n + 4 + (n^3 - 3*n^2 + 2*n)*x)*v[i - 1])/(n^2 - 3*n + 2)); 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-f3e0631eb8740f76","steps":["basis A008288"],"edges":["e-393d995dcff5ebfb"],"text":"A358735 \u2192 basis A008288"},"headline":"A358735 \u2192 basis A008288","chain":null}