{"id":"n-9c84183b4d2739be","revision":0,"first_index":1,"fingerprint":"f6-8d3b69a18d4677d2","name":"mobius_x_over_x_plus_one of basis_map_A013611 of Triangle read by rows: T(n,k) are the coefficients of Charlier polynomials: A046716 transposed, for 0 <= k <= n.","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"],["16","140","376","316"],["65","724","2822","4628","2721"],["326","4306","21484","51156","58574","25946"],["1957","29458","176079","539436","898739","775586","271801"],["13700","229400","1579596","5837856","12568924","15824056","10818644","3105936"],["109601","2009096","15539900","66628088","173936614","284025016","284028540","159361480","38474561"],["986410","19570974","167109576","810027896","2465565516","4900964676","6376350184","5245476120","2479562202","513796366"],["9864101","210022182","1954760709","10519497640","36367044906","84607357188","134420185490","144242497896","100186291473","40718397638","7360674441"],["108505112","2462653860","24745104080","145871962804","562189192272","1490739768712","2780321490368","3652641530952","3316646817752","1984417466324","704750906160","112632827396"],["1302061345","31331832860","337320107746","2155972832172","9135314057359","27090362870840","57746734050396","89275212751928","99457308456655","77941754700716","40818037334498","12835154949148","1833790646881"],["16926797486","429926795242","4929107496980","33881483691948","156157769118602","510568851831502","1220192925739896","2160845900069320","2838529131487586","2734171886436454","1878521315417140","872338216036556","245545659409574","31656637715106"],["236975164805","6329343658714","76888505711755","564670014375028","2806915189143205","10007809861095478","26429886010854323","52582186521691544","79269200989489079","90184921198053542","76277020456046121","46535609437328820","19369423203402399","4925681523350666","577636838177561"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","8","76","848","10960","161792","2693056","49978112","1024112896","22979409920","560596593664","14777528225792","418659596652544","12687785738043392","409614416067543040"],"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"],["2","6"]],"recurrence":{"denominator":{"coeffs":[["1","1"]]},"terms":[{"coeff":{"coeffs":[["2","6","6","2"],["-1","-3","-3","-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","4","4"],["1","2","1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":184838,"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-b72b98d42e22ec6d"]},{"id":184839,"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-b72b98d42e22ec6d"]},{"id":184840,"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-b72b98d42e22ec6d"]}],"region_exact":{},"incoming_edges":[{"id":"e-b72b98d42e22ec6d","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-9c84183b4d2739be","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":"2204f00cf1809011e4710d47b57799d7c3d7743f5dd109974eb23b680600e1e1","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-735a959cc5ec274b","role":"sequence","name":"basis_map_A013611 of Triangle read by rows: T(n,k) are the coefficients of Charlier polynomials: A046716 transposed, for 0 <= k <= n.","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":[],"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:     16,      140,       376,       316;\n   5:     65,      724,      2822,      4628,       2721;\n   6:    326,     4306,     21484,     51156,      58574,      25946;\n   7:   1957,    29458,    176079,    539436,     898739,     775586,     271801;\n   8:  13700,   229400,   1579596,   5837856,   12568924,   15824056,   10818644,    3105936;\n   9: 109601,  2009096,  15539900,  66628088,  173936614,  284025016,  284028540,  159361480,   38474561;\n  10: 986410, 19570974, 167109576, 810027896, 2465565516, 4900964676, 6376350184, 5245476120, 2479562202, 513796366;\n","recurrence_sentence":"The row polynomials satisfy (1+x)*P(n,x) = -(n-2+(3*n-6)*x+(3*n-6)*x^2+(n-2)*x^3)*P(n-2,x) + (n+(2*n+4)*x+(n+4)*x^2)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2 + 6*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 2 + 6*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n - 2 + (3*n - 6)*x + (3*n - 6)*x^2 + (n - 2)*x^3)*P[n - 2, x] + (n + (2*n + 4)*x + (n + 4)*x^2)*P[n - 1, x])/(1 + x)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + 6*x; for(n = 3, N, my(i = n); v[i] = (-(n - 2 + (3*n - 6)*x + (3*n - 6)*x^2 + (n - 2)*x^3)*v[i - 2] + (n + (2*n + 4)*x + (n + 4)*x^2)*v[i - 1])/(1 + x)); 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-3272d5bc697b9cb4","steps":["basis A013611","x \u21a6 x/(x+1)"],"edges":["e-91983714a64825b5","e-b72b98d42e22ec6d"],"text":"A094816 \u2192 basis A013611 \u2192 x \u21a6 x/(x+1)"},"headline":"A094816 \u2192 basis A013611 \u2192 x \u21a6 x/(x+1)","chain":null}
