{"id":"n-5bcd698d4270d7b8","revision":0,"first_index":0,"fingerprint":"f6-d769fcdab31c02d6","name":"mobius_x_over_x_plus_one of Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + ((n+1)*(n+2)/2)*T(n-2, k-1), read by rows.","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"],["1","2"],["1","10","10"],["1","22","60","40"],["1","39","269","460","230"],["1","62","830","2720","3250","1300"],["1","92","2074","13004","29102","27120","9040"],["1","130","4526","49264","182910","300244","227080","64880"],["1","177","8971","155786","959948","2560834","3357558","2146240","536560"],["1","234","16530","429808","4229970","17250300","35052696","37864176","20886840","4641520"],["1","302","28746","1066636","15963548","99348684","301924908","498583440","459865860","223480000","44696000"],["1","382","47680","2431720","52911184","494738464","2176083336","5182066944","7144548756","5725790968","2482868080","451430560"],["1","475","76017","5170448","157454386","2150307576","13658973376","46050130488","90354942312","107233774780","76118923276","29821182720","4970197120"],["1","582","117182","10368992","428132282","8276221268","75462801568","353804366240","955060982688","1582238307904","1641972143856","1043968229312","372713919760","57340603040"],["1","704","175466","19782396","1078446066","28647465912","368946679544","2401843683616","8827762178848","19860969009280","28517094810704","26330236283744","15173463097904","4977734023360","711104860480"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","3","21","123","999","8163","80433","829035","9726075","120372075","1644958125","23712918075","370521132975","6091276194675","107198900498025"],"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"],["1","2"]],"recurrence":{"denominator":{"coeffs":[["2","2"]]},"terms":[{"coeff":{"coeffs":[["0","2","4","2"],["0","3","6","3"],["0","1","2","1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["2","6","4"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":180206,"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-c3fe591ab7b2975f"]},{"id":180207,"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-c3fe591ab7b2975f"]},{"id":180208,"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-c3fe591ab7b2975f"]}],"region_exact":{},"incoming_edges":[{"id":"e-c3fe591ab7b2975f","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-5bcd698d4270d7b8","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":"af0943b9a8a7e196dafce0e1a95e882f3406a79b6f4f03b8b1ab3eaf3b995111","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-8c29b5d8d22f0df0","role":"sequence","name":"Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + ((n+1)*(n+2)/2)*T(n-2, k-1), read by rows.","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-38c623a07ea4f25f","kind":"fingerprint_related","node_a_id":"n-a9c55a824264d0f7","node_b_id":"n-5bcd698d4270d7b8","fingerprint":"f6-d769fcdab31c02d6","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-d769fcdab31c02d6"},"other_id":"n-a9c55a824264d0f7","other":{"id":"n-a9c55a824264d0f7","name":"basis_map_A007318 of Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + ((n+1)*(n+2)/2)*T(n-2, k-1), read by rows.","first_index":0,"fingerprint":"f6-d769fcdab31c02d6","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","1"],["10","10","1"]],"card_rows":[["1"],["2","1"],["10","10","1"],["40","60","22","1"]],"row_sums_preview":["1","3","21","123","999","8163","80433","829035"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-8c29b5d8d22f0df0","steps":["x \u21a6 x+1"],"edges":["e-13107c3947273d6e"],"text":"A154227 \u2192 x \u21a6 x+1"},"headline":"A154227 \u2192 x \u21a6 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: 1;\n  1: 1,   2;\n  2: 1,  10,    10;\n  3: 1,  22,    60,     40;\n  4: 1,  39,   269,    460,     230;\n  5: 1,  62,   830,   2720,    3250,     1300;\n  6: 1,  92,  2074,  13004,   29102,    27120,     9040;\n  7: 1, 130,  4526,  49264,  182910,   300244,   227080,    64880;\n  8: 1, 177,  8971, 155786,  959948,  2560834,  3357558,  2146240,   536560;\n  9: 1, 234, 16530, 429808, 4229970, 17250300, 35052696, 37864176, 20886840, 4641520;\n","recurrence_sentence":"The row polynomials satisfy (2+2*x)*P(n,x) = ((n^2+3*n+2)*x+(2*n^2+6*n+4)*x^2+(n^2+3*n+2)*x^3)*P(n-2,x) + (2+6*x+4*x^2)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 2*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 2*x;\nP[n_, x_] := P[n, x] = Expand[Together[(((n^2 + 3*n + 2)*x + (2*n^2 + 6*n + 4)*x^2 + (n^2 + 3*n + 2)*x^3)*P[n - 2, x] + (2 + 6*x + 4*x^2)*P[n - 1, x])/(2 + 2*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (((n^2 + 3*n + 2)*x + (2*n^2 + 6*n + 4)*x^2 + (n^2 + 3*n + 2)*x^3)*v[i - 2] + (2 + 6*x + 4*x^2)*v[i - 1])/(2 + 2*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-8c29b5d8d22f0df0","steps":["x \u21a6 x/(x+1)"],"edges":["e-c3fe591ab7b2975f"],"text":"A154227 \u2192 x \u21a6 x/(x+1)"},"headline":"A154227 \u2192 x \u21a6 x/(x+1)","chain":null}
