{"id":"n-70b12dfbbd98514b","revision":0,"first_index":1,"fingerprint":"f6-07ec67562a37d493","name":"mobius_x_over_x_plus_one of basis_map_A013611 of Square array of Delannoy numbers D(i,j) (i >= 0, j >= 0) read by antidiagonals.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["2","6"],["5","30","41"],["12","108","308","276"],["29","348","1510","2796","1861"],["70","1050","6124","17316","23694","12546"],["169","3042","22291","84972","177471","192402","84581"],["408","8568","75608","363000","1022984","1690920","1517320","570216"],["985","23640","244012","1413576","5022726","11201832","15306508","11713848","3844201"],["2378","64206","758920","5150712","22105932","62183844","114601768","133395480","88979386","25916286"],["5741","172230","2293853","17850792","89840378","305409348","709937730","1113963624","1128991481","667343334","174718721"],["13860","457380","6777132","59489964","343590312","1370433960","3850566040","7619478936","10404125812","9335405556","4953892092","1177893756"],["33461","1204596","19654482","192117780","1252534803","5736115368","18915331548","45242990856","77887587115","94106676324","75743953650","36464169060","7940956141"],["80782","3150498","56127252","604602108","4392192042","22716865206","86038476664","241553015976","502606948994","765396430446","829114919732","604970334108","266503749638","53535205626"],["195025","8191050","158206855","1862035380","14915510625","85998795750","367972842335","1186752448920","2898454183835","5334081708150","7280430369309","7146091459188","4768275242675","1936059491226","360916014461"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","8","76","704","6544","60800","564928","5249024","48771328","453158912","4210527232","39122124800","363503325184","3377492099072","31381976694784"],"nonn":true,"definitions":[{"position":0,"source":"derived:mobius_x_over_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["-2","-6","-6","-2"],["2","6","6","2"]]},"terms":[{"coeff":{"coeffs":[["58","468","1510","2512","2286","1084","210"],["-45","-354","-1149","-1948","-1815","-882","-175"],["9","66","209","356","339","170","35"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["20","236","1082","2578","3512","2768","1178","210"],["-6","-70","-322","-778","-1082","-874","-382","-70"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["1","16","102","340","660","776","546","212","35"]]},"deriv_order":2,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":182399,"position":0,"claim":"real_rooted","from_index":1,"region_key":"[-1, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in [-1, 0]","region_exact":null,"edges":["e-301e8e82cf7b951d"]},{"id":182400,"position":1,"claim":"one_signed_coefficients","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":1},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 1: the coefficients of P_n have one sign","region_exact":null,"edges":["e-301e8e82cf7b951d"]},{"id":182401,"position":2,"claim":"hadamard_kernel","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":1},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-301e8e82cf7b951d"]}],"region_exact":{},"incoming_edges":[{"id":"e-301e8e82cf7b951d","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"mobius\",\"a\":\"1\",\"b\":\"0\",\"c\":\"1\",\"e\":\"1\",\"schedule\":{\"first_index\":1,\"first_degree\":0,\"step\":1}}"},"operation_detail":"mobius, shift 0","conclusion_id":"n-70b12dfbbd98514b","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":"40e65bb9f6bc0f07f499d50a5ea48c93fbc24fdc9da143a8da85b48d99151eab","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-ccd84167c908ea61","role":"sequence","name":"basis_map_A013611 of Square array of Delannoy numbers D(i,j) (i >= 0, j >= 0) read by antidiagonals.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-57cda5ca31e496d9","kind":"fingerprint_related","node_a_id":"n-70b12dfbbd98514b","node_b_id":"n-800db8dcc26dde3d","fingerprint":"f6-07ec67562a37d493","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-07ec67562a37d493"},"other_id":"n-800db8dcc26dde3d","other":{"id":"n-800db8dcc26dde3d","name":"mobius_x_over_x_plus_one of basis_map_A013611 of delannoy","first_index":0,"fingerprint":"f6-07ec67562a37d493","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","6"],["5","30","41"]],"card_rows":[["1"],["2","6"],["5","30","41"],["12","108","308","276"]],"row_sums_preview":["1","8","76","704","6544","60800","564928","5249024"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-e00a1cb428776eaa","steps":["basis A013611","x \u21a6 x/(x+1)"],"edges":["e-e469b77fb7f7bdb9","e-4fccc095945fd6e9"],"text":"delannoy \u2192 basis A013611 \u2192 x \u21a6 x/(x+1)"},"headline":"delannoy \u2192 basis A013611 \u2192 x \u21a6 x/(x+1)","chain":null}}],"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:    1;\n   2:    2,     6;\n   3:    5,    30,     41;\n   4:   12,   108,    308,     276;\n   5:   29,   348,   1510,    2796,     1861;\n   6:   70,  1050,   6124,   17316,    23694,    12546;\n   7:  169,  3042,  22291,   84972,   177471,   192402,     84581;\n   8:  408,  8568,  75608,  363000,  1022984,  1690920,   1517320,    570216;\n   9:  985, 23640, 244012, 1413576,  5022726, 11201832,  15306508,  11713848,  3844201;\n  10: 2378, 64206, 758920, 5150712, 22105932, 62183844, 114601768, 133395480, 88979386, 25916286;\n","recurrence_sentence":"The row polynomials satisfy (2*n-2+(6*n-6)*x+(6*n-6)*x^2+(2*n-2)*x^3)*P(n,x) = (9*n^2-45*n+58+(66*n^2-354*n+468)*x+(209*n^2-1149*n+1510)*x^2+(356*n^2-1948*n+2512)*x^3+(339*n^2-1815*n+2286)*x^4+(170*n^2-882*n+1084)*x^5+(35*n^2-175*n+210)*x^6)*P(n-1,x) - (6*n-20+(70*n-236)*x+(322*n-1082)*x^2+(778*n-2578)*x^3+(1082*n-3512)*x^4+(874*n-2768)*x^5+(382*n-1178)*x^6+(70*n-210)*x^7)*d/dx P(n-1,x) + (1+16*x+102*x^2+340*x^3+660*x^4+776*x^5+546*x^6+212*x^7+35*x^8)*d^2/dx^2 P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((9*n^2 - 45*n + 58 + (66*n^2 - 354*n + 468)*x + (209*n^2 - 1149*n + 1510)*x^2 + (356*n^2 - 1948*n + 2512)*x^3 + (339*n^2 - 1815*n + 2286)*x^4 + (170*n^2 - 882*n + 1084)*x^5 + (35*n^2 - 175*n + 210)*x^6)*P[n - 1, x] - (6*n - 20 + (70*n - 236)*x + (322*n - 1082)*x^2 + (778*n - 2578)*x^3 + (1082*n - 3512)*x^4 + (874*n - 2768)*x^5 + (382*n - 1178)*x^6 + (70*n - 210)*x^7)*D[P[n - 1, x], x] + (1 + 16*x + 102*x^2 + 340*x^3 + 660*x^4 + 776*x^5 + 546*x^6 + 212*x^7 + 35*x^8)*D[P[n - 1, x], {x, 2}])/(2*n - 2 + (6*n - 6)*x + (6*n - 6)*x^2 + (2*n - 2)*x^3)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((9*n^2 - 45*n + 58 + (66*n^2 - 354*n + 468)*x + (209*n^2 - 1149*n + 1510)*x^2 + (356*n^2 - 1948*n + 2512)*x^3 + (339*n^2 - 1815*n + 2286)*x^4 + (170*n^2 - 882*n + 1084)*x^5 + (35*n^2 - 175*n + 210)*x^6)*v[i - 1] - (6*n - 20 + (70*n - 236)*x + (322*n - 1082)*x^2 + (778*n - 2578)*x^3 + (1082*n - 3512)*x^4 + (874*n - 2768)*x^5 + (382*n - 1178)*x^6 + (70*n - 210)*x^7)*deriv(v[i - 1], x) + (1 + 16*x + 102*x^2 + 340*x^3 + 660*x^4 + 776*x^5 + 546*x^6 + 212*x^7 + 35*x^8)*deriv(deriv(v[i - 1], x), x))/(2*n - 2 + (6*n - 6)*x + (6*n - 6)*x^2 + (2*n - 2)*x^3)); 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-811c78acf96cbc19","steps":["basis A013611","x \u21a6 x/(x+1)"],"edges":["e-8e7f1943c60a0009","e-301e8e82cf7b951d"],"text":"A008288 \u2192 basis A013611 \u2192 x \u21a6 x/(x+1)"},"headline":"A008288 \u2192 basis A013611 \u2192 x \u21a6 x/(x+1)","chain":null}
