{"id":"n-5fd44c17d91e2308","revision":0,"first_index":0,"fingerprint":"f6-06597c4b2866cc52","name":"mobius_x_over_x_plus_one of basis_map_A007318 of Triangle read by rows: T(n,k) is the number of double rise-bicolored Dyck paths (double rises come in two colors; also called marked Dyck paths) of semilength n and having k double rises (0 <= k <= n-1).","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","5"],["11","36","29"],["45","219","351","185"],["197","1272","3050","3216","1257"],["903","7265","23190","36690","28755","8925"],["4279","41220","164295","346680","408285","254340","65445"],["20793","233303","1115037","2941715","4625355","4332965","2238495","491825"],["103049","1319760","7354676","23288160","45816974","57336720","44560292","19658304","3768209"],["518859","7468317","47543436","175660548","415041354","650211030","675273564","448217748","172504323","29324405"],["2646723","42295140","302800305","1278752400","3527193390","6638845752","8633832690","7659498960","4435482495","1513715460","231153133"],["13648869","239763315","1906687035","9059583885","28574147250","62807196606","98160199830","109067584890","84422918745","43349519455","13287845439","1841801065"],["71039373","1360601928","11899116702","62826168240","223023929535","560706399792","1023613317012","1367054333568","1325435738715","909742549320","419551261998","116714415888","14810069497"],["372693519","7729279545","73726162146","428227298070","1689691266549","4782399256635","9988132946220","15583978411716","18163025809929","15616335178015","9626706773970","4028847564742","1025890986003","120029657805"],["1968801519","43953686532","454112160411","2877625420488","12494009346819","39314891099484","92454974154543","165054325323888","224764778403165","232318994071164","179400126466561","100356253515336","38440728729585","9024125069988","979470140661"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","8","76","800","8992","105728","1284544","15999488","203206144","2621763584","34266216448","452730896384","6036808941568","81135093284864","1097980336390144"],"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":[["2","4","2"],["1","2","1"]]},"terms":[{"coeff":{"coeffs":[["0","-8","-28","-32","-12"],["9","41","67","47","12"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-2","-16","-48","-68","-46","-12"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":182358,"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-5de9d9922d645c0a"]},{"id":182359,"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-5de9d9922d645c0a"]},{"id":182360,"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-5de9d9922d645c0a"]}],"region_exact":{},"incoming_edges":[{"id":"e-5de9d9922d645c0a","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-5fd44c17d91e2308","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":"0e4ddb5d5d29c4f2aea97603457e0fb20b8ec491f25361c9f7f00bdd2875a0dd","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-b6cda967a092feb1","role":"sequence","name":"basis_map_A007318 of Triangle read by rows: T(n,k) is the number of double rise-bicolored Dyck paths (double rises come in two colors; also called marked Dyck paths) of semilength n and having k double rises (0 <= k <= n-1).","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:      3,       5;\n  2:     11,      36,       29;\n  3:     45,     219,      351,       185;\n  4:    197,    1272,     3050,      3216,      1257;\n  5:    903,    7265,    23190,     36690,     28755,      8925;\n  6:   4279,   41220,   164295,    346680,    408285,    254340,     65445;\n  7:  20793,  233303,  1115037,   2941715,   4625355,   4332965,   2238495,    491825;\n  8: 103049, 1319760,  7354676,  23288160,  45816974,  57336720,  44560292,  19658304,   3768209;\n  9: 518859, 7468317, 47543436, 175660548, 415041354, 650211030, 675273564, 448217748, 172504323, 29324405;\n","recurrence_sentence":"The row polynomials satisfy (n+2+(2*n+4)*x+(n+2)*x^2)*P(n,x) = (9*n+(41*n-8)*x+(67*n-28)*x^2+(47*n-32)*x^3+(12*n-12)*x^4)*P(n-1,x) - (2+16*x+48*x^2+68*x^3+46*x^4+12*x^5)*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[((9*n + (41*n - 8)*x + (67*n - 28)*x^2 + (47*n - 32)*x^3 + (12*n - 12)*x^4)*P[n - 1, x] - (2 + 16*x + 48*x^2 + 68*x^3 + 46*x^4 + 12*x^5)*D[P[n - 1, x], x])/(n + 2 + (2*n + 4)*x + (n + 2)*x^2)]];\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] = ((9*n + (41*n - 8)*x + (67*n - 28)*x^2 + (47*n - 32)*x^3 + (12*n - 12)*x^4)*v[i - 1] - (2 + 16*x + 48*x^2 + 68*x^3 + 46*x^4 + 12*x^5)*deriv(v[i - 1], x))/(n + 2 + (2*n + 4)*x + (n + 2)*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-8ce61d302cec295b","steps":["basis A007318","x \u21a6 x/(x+1)"],"edges":["e-6804282af6df94ec","e-5de9d9922d645c0a"],"text":"A114687 \u2192 basis A007318 \u2192 x \u21a6 x/(x+1)"},"headline":"A114687 \u2192 basis A007318 \u2192 x \u21a6 x/(x+1)","chain":null}