{"id":"n-3cb0df9d7da51f9d","revision":0,"first_index":0,"fingerprint":"f6-8ebc68b94e36a726","name":"mobius_x_over_x_plus_one of shift_x_plus_one of Triangle read by rows: T(n,k) = k*(k+1)*binomial(n,k)/2 (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":true,"rows":[["1"],["5","8"],["18","57","45"],["56","264","414","216"],["160","1000","2340","2430","945"],["432","3360","10440","16200","12555","3888"],["1120","10416","40320","83160","96390","59535","15309"],["2816","30464","141120","362880","559440","517104","265356","58320"],["6912","85248","459648","1415232","2721600","3347568","2571912","1128492","216513"],["16640","230400","1416960","5080320","11702880","17962560","18370800","12072240","4625505","787320"],["39424","605440","4181760","17107200","45904320","84424032","107775360","94303440","54128250","18403605","2814669"],["92160","1554432","11911680","54743040","167650560","359251200","549654336","600462720","459007560","233834040","71449290","9920232"],["212992","3913728","32947200","168030720","578223360","1414423296","2521943424","3302544960","3152429280","2139148440","979504812","271743498","34543665"],["487424","9691136","88897536","498161664","1902700800","5230697472","10648205568","16252524288","18599332752","15762146400","9614909304","3997853496","1015583751","119042784"],["1105920","23654400","234823680","1434101760","6019453440","18369711360","42032390400","73256451840","97725307680","99301522320","75658302720","41912980200","15959173230","3738687435","406552365"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","13","120","950","6875","46875","306250","1937500","11953125","72265625","429687500","2519531250","14599609375","83740234375","476074218750"],"nonn":true,"definitions":[{"position":0,"source":"derived:mobius_x_over_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["5","8"],["18","57","45"]],"recurrence":{"denominator":{"coeffs":[["1","3","3","1"]]},"terms":[{"coeff":{"coeffs":[["8","60","186","305","279","135","27"]]},"deriv_order":0,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["-12","-72","-171","-201","-117","-27"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["6","27","45","33","9"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":177360,"position":0,"claim":"real_rooted","from_index":0,"region_key":"[-1, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in [-1, 0]","region_exact":null,"edges":["e-567ecf0263f0f647"]},{"id":177361,"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-567ecf0263f0f647"]},{"id":177362,"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-567ecf0263f0f647"]}],"region_exact":{},"incoming_edges":[{"id":"e-567ecf0263f0f647","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"mobius\",\"a\":\"1\",\"b\":\"0\",\"c\":\"1\",\"e\":\"1\",\"schedule\":{\"first_index\":0,\"first_degree\":0,\"step\":1}}"},"operation_detail":"mobius, shift 0","conclusion_id":"n-3cb0df9d7da51f9d","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":"fTransform","lean_declarations":["RealRooted.splits_fPolynomial"],"certificate_hash":"90b5154b68a993e5851fd3f4d246e04b1e0256f2df6fa04efd379645cc736bd2","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-ff3909e492eb580d","role":"sequence","name":"shift_x_plus_one of Triangle read by rows: T(n,k) = k*(k+1)*binomial(n,k)/2 (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":[],"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:     5,      8;\n  2:    18,     57,      45;\n  3:    56,    264,     414,     216;\n  4:   160,   1000,    2340,    2430,      945;\n  5:   432,   3360,   10440,   16200,    12555,     3888;\n  6:  1120,  10416,   40320,   83160,    96390,    59535,    15309;\n  7:  2816,  30464,  141120,  362880,   559440,   517104,   265356,    58320;\n  8:  6912,  85248,  459648, 1415232,  2721600,  3347568,  2571912,  1128492,  216513;\n  9: 16640, 230400, 1416960, 5080320, 11702880, 17962560, 18370800, 12072240, 4625505, 787320;\n","recurrence_sentence":"The row polynomials satisfy (1+3*x+3*x^2+x^3)*P(n,x) = (8+60*x+186*x^2+305*x^3+279*x^4+135*x^5+27*x^6)*P(n-3,x) - (12+72*x+171*x^2+201*x^3+117*x^4+27*x^5)*P(n-2,x) + (6+27*x+45*x^2+33*x^3+9*x^4)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 5 + 8*x, P(2,x) = 18 + 57*x + 45*x^2.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 5 + 8*x;\nP[2, x_] := 18 + 57*x + 45*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((8 + 60*x + 186*x^2 + 305*x^3 + 279*x^4 + 135*x^5 + 27*x^6)*P[n - 3, x] - (12 + 72*x + 171*x^2 + 201*x^3 + 117*x^4 + 27*x^5)*P[n - 2, x] + (6 + 27*x + 45*x^2 + 33*x^3 + 9*x^4)*P[n - 1, x])/(1 + 3*x + 3*x^2 + x^3)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 5 + 8*x; v[3] = 18 + 57*x + 45*x^2; for(n = 3, N, my(i = n + 1); v[i] = ((8 + 60*x + 186*x^2 + 305*x^3 + 279*x^4 + 135*x^5 + 27*x^6)*v[i - 3] - (12 + 72*x + 171*x^2 + 201*x^3 + 117*x^4 + 27*x^5)*v[i - 2] + (6 + 27*x + 45*x^2 + 33*x^3 + 9*x^4)*v[i - 1])/(1 + 3*x + 3*x^2 + x^3)); 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-787407680a56b31c","steps":["basis A007318","x \u21a6 x/(x+1)"],"edges":["e-70e46ddc2d19c133","e-567ecf0263f0f647"],"text":"A124932 \u2192 basis A007318 \u2192 x \u21a6 x/(x+1)"},"headline":"A124932 \u2192 basis A007318 \u2192 x \u21a6 x/(x+1)","chain":null}
