{"id":"n-21cb97761b658085","revision":0,"first_index":0,"fingerprint":"f6-2b67a3b9cedf1278","name":"mobius_x_over_x_plus_one of basis_map_A007318 of Triangle of coefficients in expansion of (1+x)*(1+3x)*(1+5x)*(1+7x)*...*(1+(2n-1)x).","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":true,"rows":[["1"],["2","3"],["8","26","21"],["48","244","412","231"],["384","2672","6956","8028","3465"],["3840","34016","120328","212444","187182","65835"],["46080","496512","2226304","5316872","7132396","5095206","1514205"],["645120","8195328","44574080","134546416","243409088","263907576","158769432","40883535"],["10321920","151123968","967240448","3534539136","8065484304","11768202944","10721445768","5575988952","1267389585"],["185794560","3081498624","22699666944","97475120128","268887587232","494119603632","604873126864","475618403016","217972625850","44358635475"],["3715891200","68875960320","574171785216","2834789413376","9179281429632","20369007974688","31368127078928","33102420008016","22908570234624","9388105117650","1729986783525"],["81749606400","1675054448640","15593445568512","87054753858560","323840136227072","842827276917312","1565966138648000","2077082704570256","1927392605506416","1191606832677132","441748229296500","74389431691575"],["1961990553600","44043538268160","452970252730368","2822206034325504","11863736700802048","35448341048687872","77196069342665664","123450393426142144","143880309646956016","119186016443052720","66607478638941204","22547513137533300","3496303289504025"],["51011754393600","1245193513205760","14023447022665728","96478839781711872","452389661971453952","1526707439006789120","3814963196392388736","7146709765555644608","10036858115554105760","10436732219514127536","7810281283208160024","3983216752161867204","1240827055541302950","178311467764705275"],["1428329123020800","37671064861409280","461142159860957184","3472697100134547456","17973246723194863616","67629239700620062720","190787878644360286208","409930849237139429504","674101064341075414720","844255698501871387808","792708148003105495152","541095539636981083032","253820078924059178820","73238209152183409950","9807130727058790125"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","5","55","935","21505","623645","21827575","894930575","42061737025","2229272062325","131527051677175","8549258359016375","606997343490162625","46738795448742522125","3879320022245629336375"],"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":[["0","-1","-1"],["2","6","4"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":187916,"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-3d893738d753a273"]},{"id":187917,"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-3d893738d753a273"]},{"id":187918,"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-3d893738d753a273"]}],"region_exact":{},"incoming_edges":[{"id":"e-3d893738d753a273","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-21cb97761b658085","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":"2dcb0f5a008a5e5531399bc19afd9ba66220292848571845767c2f1763ee738c","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-261c6497576cc5f7","role":"sequence","name":"basis_map_A007318 of Triangle of coefficients in expansion of (1+x)*(1+3x)*(1+5x)*(1+7x)*...*(1+(2n-1)x).","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-52531a2676ee5d99","kind":"fingerprint_related","node_a_id":"n-45bcdc2be26ae0cd","node_b_id":"n-21cb97761b658085","fingerprint":"f6-2b67a3b9cedf1278","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2b67a3b9cedf1278"},"other_id":"n-45bcdc2be26ae0cd","other":{"id":"n-45bcdc2be26ae0cd","name":"shift_x_plus_one of basis_map_A013609 of Triangle T(n, k) read by rows: T(n, k) = Sum_{j=0..n} binomial(j, n-k) * |Stirling1(n, n-j)|.","first_index":0,"fingerprint":"f6-2b67a3b9cedf1278","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["3","2"],["21","26","8"]],"card_rows":[["1"],["3","2"],["21","26","8"],["231","412","244","48"]],"row_sums_preview":["1","5","55","935","21505","623645","21827575","894930575"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-849de7332df28657","steps":["basis A013609","x \u21a6 x+1"],"edges":["e-0b105283f866e3e7","e-9fad52f644c1f0a6"],"text":"A088996 \u2192 basis A013609 \u2192 x \u21a6 x+1"},"headline":"A088996 \u2192 basis A013609 \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:         2,          3;\n  2:         8,         26,          21;\n  3:        48,        244,         412,         231;\n  4:       384,       2672,        6956,        8028,         3465;\n  5:      3840,      34016,      120328,      212444,       187182,        65835;\n  6:     46080,     496512,     2226304,     5316872,      7132396,      5095206,      1514205;\n  7:    645120,    8195328,    44574080,   134546416,    243409088,    263907576,    158769432,     40883535;\n  8:  10321920,  151123968,   967240448,  3534539136,   8065484304,  11768202944,  10721445768,   5575988952,   1267389585;\n  9: 185794560, 3081498624, 22699666944, 97475120128, 268887587232, 494119603632, 604873126864, 475618403016, 217972625850, 44358635475;\n","recurrence_sentence":"The row polynomials satisfy (1+x)*P(n,x) = (2*n+(6*n-1)*x+(4*n-1)*x^2)*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[((2*n + (6*n - 1)*x + (4*n - 1)*x^2)*P[n - 1, 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] = ((2*n + (6*n - 1)*x + (4*n - 1)*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 - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-2b477f2aa77e3efc","steps":["basis A007318","x \u21a6 x/(x+1)"],"edges":["e-07dfe9939e4be29b","e-3d893738d753a273"],"text":"A109692 \u2192 basis A007318 \u2192 x \u21a6 x/(x+1)"},"headline":"A109692 \u2192 basis A007318 \u2192 x \u21a6 x/(x+1)","chain":null}