{"id":"n-96631c9ffb8654db","revision":0,"first_index":0,"fingerprint":"f6-c5be635e87f1100d","name":"mobius_x_over_x_plus_one of theta_plus_one of Triangle read by rows: T(n,k) = |s(n,n+1-k)|, where s(n,k) are the signed Stirling numbers of the first kind A008276 (1 <= k <= n; in other words, the unsigned Stirling numbers of the first kind in reverse order).","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","3"],["1","8","13"],["1","15","60","70"],["1","24","171","474","446"],["1","35","385","1855","4060","3276"],["1","48","750","5480","20525","37960","27252"],["1","63","1323","13545","76020","237888","387128","253296"],["1","80","2170","29540","230125","1069460","2919000","4291560","2602224"],["1","99","3366","58674","603141","3862551","15545684","38062596","51491088","29288160"],["1","120","4995","108360","1417395","11905992","65645405","235622760","527717004","665688528","358457760"],["1","143","7150","188760","3056625","32507475","233937704","1144682110","3739828664","7771492872","9233617536","4740577920"],["1","168","9933","313390","6148065","80599596","731298491","4650991422","20628418734","62268024896","121352444136","136866901248","67375532160"],["1","195","13455","499785","11672375","184703805","2058367025","16445970255","94467759672","385709970360","1087910858320","2005073843760","2159996746752","1024208720640"],["1","224","17836","770224","21107086","396347952","5314737428","51993424912","374025717065","1971588482864","7497985430936","19935414180864","34978276586448","36172182908160","16583626886400"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","4","22","146","1116","9612","92016","969264","11144160","138915360","1866568320","26899896960","413960682240","6776069126400","117570826598400"],"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","3"]],"recurrence":{"denominator":{"coeffs":[["-1","-2","-1"],["1","2","1"]]},"terms":[{"coeff":{"coeffs":[["0","0","0","0","0"],["-1","-2","-1","0","0"],["0","-2","-4","-2","0"],["0","0","-1","-2","-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-1","-3","-3","-1"],["2","4","2","0"],["0","2","4","2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":181335,"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-36d47604395285d1"]},{"id":181336,"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-36d47604395285d1"]},{"id":181337,"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-36d47604395285d1"]}],"region_exact":{},"incoming_edges":[{"id":"e-36d47604395285d1","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-96631c9ffb8654db","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":"683f771f66e0aa4f8b468054ddd7067e90dac1443effc5282ae384a206f8d860","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-994dbf40e5f5e735","role":"sequence","name":"theta_plus_one of Triangle read by rows: T(n,k) = |s(n,n+1-k)|, where s(n,k) are the signed Stirling numbers of the first kind A008276 (1 <= k <= n; in other words, the unsigned Stirling numbers of the first kind in reverse order).","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: 1,  3;\n  2: 1,  8,   13;\n  3: 1, 15,   60,    70;\n  4: 1, 24,  171,   474,    446;\n  5: 1, 35,  385,  1855,   4060,    3276;\n  6: 1, 48,  750,  5480,  20525,   37960,    27252;\n  7: 1, 63, 1323, 13545,  76020,  237888,   387128,   253296;\n  8: 1, 80, 2170, 29540, 230125, 1069460,  2919000,  4291560,  2602224;\n  9: 1, 99, 3366, 58674, 603141, 3862551, 15545684, 38062596, 51491088, 29288160;\n","recurrence_sentence":"The row polynomials satisfy (n-1+(2*n-2)*x+(n-1)*x^2)*P(n,x) = -(n+(2*n^2+2*n)*x+(n^3+4*n^2+n)*x^2+(2*n^3+2*n^2)*x^3+n^3*x^4)*P(n-2,x) + (2*n-1+(2*n^2+4*n-3)*x+(4*n^2+2*n-3)*x^2+(2*n^2-1)*x^3)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 3*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 3*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n + (2*n^2 + 2*n)*x + (n^3 + 4*n^2 + n)*x^2 + (2*n^3 + 2*n^2)*x^3 + n^3*x^4)*P[n - 2, x] + (2*n - 1 + (2*n^2 + 4*n - 3)*x + (4*n^2 + 2*n - 3)*x^2 + (2*n^2 - 1)*x^3)*P[n - 1, x])/(n - 1 + (2*n - 2)*x + (n - 1)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n + (2*n^2 + 2*n)*x + (n^3 + 4*n^2 + n)*x^2 + (2*n^3 + 2*n^2)*x^3 + n^3*x^4)*v[i - 2] + (2*n - 1 + (2*n^2 + 4*n - 3)*x + (4*n^2 + 2*n - 3)*x^2 + (2*n^2 - 1)*x^3)*v[i - 1])/(n - 1 + (2*n - 2)*x + (n - 1)*x^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-4d60c3dd8627c01d","steps":["\u03b8+1","x \u21a6 x/(x+1)"],"edges":["e-8baeeca6d59e95f0","e-36d47604395285d1"],"text":"A094638 \u2192 \u03b8+1 \u2192 x \u21a6 x/(x+1)"},"headline":"A094638 \u2192 \u03b8+1 \u2192 x \u21a6 x/(x+1)","chain":null}
