{"id":"n-9d0af41400c21a44","revision":0,"first_index":0,"fingerprint":"f6-b373e0a412d5b427","name":"mobius_x_over_x_plus_one of basis_map_A008277 of Triangle whose (i,j)-th entry is binomial(i,j)*2^(i-j).","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"],["3","4"],["9","25","17"],["27","118","167","77"],["81","499","1108","1061","372"],["243","1996","6214","9301","6754","1915"],["729","7741","31851","66464","75141","43996","10481"],["2187","29506","154881","423781","664439","603422","296015","60814"],["6561","111463","729590","2520531","5148011","6454447","4894772","2065431","372939"],["19683","419608","3372536","14342945","36699599","59624965","62187586","40449145","14967095","2409837"],["59049","1580017","15426709","79328270","247632386","501040847","674315104","601304692","342117592","112657799","16360786"],["177147","5965774","70222987","430896049","1610013030","3948293658","6587205008","7553733309","5876470632","2969155432","880348622","116393205"],["531441","22623067","319337448","2315049593","10207188278","29749193902","59859634836","84783596182","84581285075","58303167636","26475829942","7135726092","865549453"],["1594323","86240740","1454571546","12364094029","63626844280","217158104879","516577595104","879030143850","1079937903901","952370311322","588892325306","242700621684","59931747199","6713065156"],["4782969","330614053","6648446351","65876817564","392295340867","1549997054462","4293714862295","8602769384780","12658424496433","13713828200071","10827243511371","6066143036654","2287407313959","520984007113","54190360453"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","7","51","389","3121","26423","236403","2235045","22303745","234492999","2591823251","30048874853","364618712945","4620845163319","61039858229395"],"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"]],"recurrence":{"denominator":{"coeffs":[["1","1"]]},"terms":[{"coeff":{"coeffs":[["3","8","6","1"],["0","-1","-2","-1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1","3","3","1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":178529,"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-5f0c953eb2af1537"]},{"id":178530,"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-5f0c953eb2af1537"]},{"id":178531,"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-5f0c953eb2af1537"]}],"region_exact":{},"incoming_edges":[{"id":"e-5f0c953eb2af1537","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-9d0af41400c21a44","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":"7dd941c31d856b5cef9d36d6768999624c680eb34b4cff9f436353322e3d101b","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-40f55dba6bf772b4","role":"sequence","name":"basis_map_A008277 of Triangle whose (i,j)-th entry is binomial(i,j)*2^(i-j).","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-ac9c6fbd5eab6771","kind":"fingerprint_related","node_a_id":"n-f623710345608480","node_b_id":"n-9d0af41400c21a44","fingerprint":"f6-b373e0a412d5b427","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b373e0a412d5b427"},"other_id":"n-f623710345608480","other":{"id":"n-f623710345608480","name":"shift_x_plus_one of reversal of Triangle read by rows: 3-Stirling numbers of the second kind.","first_index":1,"fingerprint":"f6-b373e0a412d5b427","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["4","3"],["17","25","9"]],"card_rows":[["1"],["4","3"],["17","25","9"],["77","167","118","27"]],"row_sums_preview":["1","7","51","389","3121","26423","236403","2235045"],"regions":["(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a2b8ff03da0c14a3","steps":["reverse","x \u21a6 x+1"],"edges":["e-0c475173369de966","e-cf2821b1bfb70d52"],"text":"A143495 \u2192 reverse \u2192 x \u21a6 x+1"},"headline":"A143495 \u2192 reverse \u2192 x \u21a6 x+1","chain":null}},{"id":"l-be38f89ddea27e62","kind":"fingerprint_related","node_a_id":"n-5fdfd6376d0f2ea7","node_b_id":"n-9d0af41400c21a44","fingerprint":"f6-b373e0a412d5b427","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b373e0a412d5b427"},"other_id":"n-5fdfd6376d0f2ea7","other":{"id":"n-5fdfd6376d0f2ea7","name":"mobius_x_over_x_plus_one of Triangle read by rows: 3-Stirling numbers of the second kind.","first_index":1,"fingerprint":"f6-b373e0a412d5b427","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["3","4"],["9","25","17"]],"card_rows":[["1"],["3","4"],["9","25","17"],["27","118","167","77"]],"row_sums_preview":["1","7","51","389","3121","26423","236403","2235045"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a2b8ff03da0c14a3","steps":["x \u21a6 x/(x+1)"],"edges":["e-76f72502b7764522"],"text":"A143495 \u2192 x \u21a6 x/(x+1)"},"headline":"A143495 \u2192 x \u21a6 x/(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:     3,      4;\n  2:     9,     25,      17;\n  3:    27,    118,     167,       77;\n  4:    81,    499,    1108,     1061,      372;\n  5:   243,   1996,    6214,     9301,     6754,     1915;\n  6:   729,   7741,   31851,    66464,    75141,    43996,    10481;\n  7:  2187,  29506,  154881,   423781,   664439,   603422,   296015,    60814;\n  8:  6561, 111463,  729590,  2520531,  5148011,  6454447,  4894772,  2065431,   372939;\n  9: 19683, 419608, 3372536, 14342945, 36699599, 59624965, 62187586, 40449145, 14967095, 2409837;\n","recurrence_sentence":"The row polynomials satisfy (1+x)*P(n,x) = (3-(n-8)*x-(2*n-6)*x^2-(n-1)*x^3)*P(n-1,x) + (x+3*x^2+3*x^3+x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((3 - (n - 8)*x - (2*n - 6)*x^2 - (n - 1)*x^3)*P[n - 1, x] + (x + 3*x^2 + 3*x^3 + x^4)*D[P[n - 1, x], x])/(1 + x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((3 - (n - 8)*x - (2*n - 6)*x^2 - (n - 1)*x^3)*v[i - 1] + (x + 3*x^2 + 3*x^3 + x^4)*deriv(v[i - 1], x))/(1 + 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-af754f07daf65b5e","steps":["basis A008277","x \u21a6 x/(x+1)"],"edges":["e-74a6e78fdd5274b8","e-5f0c953eb2af1537"],"text":"A038207 \u2192 basis A008277 \u2192 x \u21a6 x/(x+1)"},"headline":"A038207 \u2192 basis A008277 \u2192 x \u21a6 x/(x+1)","chain":null}
