{"id":"n-1a7a27d6b2026223","revision":1,"first_index":0,"fingerprint":"f6-663a3c2c5c184f3a","name":"mobius_x_over_x_plus_one of Generalized Narayana numbers, T(n, k) = 3/(n + 1)*binomial(n + 1, k + 2)*binomial(n + 1, k - 1), triangular array 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"],["3","6"],["6","27","27"],["10","75","165","110"],["15","165","594","858","429"],["21","315","1638","3822","4095","1638"],["28","546","3822","12740","21840","18564","6188"],["36","882","7938","35280","85680","115668","81396","23256"],["45","1350","15120","85680","275400","523260","581400","348840","87210"],["55","1980","26928","188496","767448","1918620","2984520","2813976","1470942","326876"],["66","2805","45441","383724","1918620","6043653","12311145","16180362","13238478","6128925","1225785"],["78","3861","73359","733590","4401540","16945929","43306263","74239308","84362850","60928725","25308855","4601610"],["91","5187","114114","1331330","9414405","43306263","134730596","288708420","426501075","426501075","275585310","103791870","17298645"],["105","6825","171990","2312310","18993975","102567465","379879500","987686700","1818241425","2356979625","2103151050","1229114250","423361575","65132550"],["120","8820","252252","3867864","36468432","227927700","987686700","3047718960","6788101320","10936385460","12618906300","10160677800","5419028160","1719499320","245642760"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","9","60","360","2061","11529","63728","350136","1918305","10499841","57479004","314905968","1727288381","9487599345","52192171968"],"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":[["0","0","0"],["6","12","6"],["1","2","1"]]},"terms":[{"coeff":{"coeffs":[["12","48","60","24"],["8","32","40","16"],["1","4","5","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","4","12","12","4"],["0","2","6","6","2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":164986,"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-8d309dafbf46f610"]},{"id":164987,"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-8d309dafbf46f610"]},{"id":164988,"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-8d309dafbf46f610"]}],"region_exact":{},"incoming_edges":[{"id":"e-8d309dafbf46f610","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-1a7a27d6b2026223","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":"5ae0a850c9f40492085f052e2ae2a20905035a6be404afceca8758098cdbced8","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-bb74f03367de3ec8","role":"sequence","name":"Generalized Narayana numbers, T(n, k) = 3/(n + 1)*binomial(n + 1, k + 2)*binomial(n + 1, k - 1), triangular array 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-3e07a180cbb9d416","kind":"fingerprint_related","node_a_id":"n-4355c823139108ab","node_b_id":"n-1a7a27d6b2026223","fingerprint":"f6-663a3c2c5c184f3a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-663a3c2c5c184f3a"},"other_id":"n-4355c823139108ab","other":{"id":"n-4355c823139108ab","name":"basis_map_A007318 of Generalized Narayana numbers, T(n, k) = 3/(n + 1)*binomial(n + 1, k + 2)*binomial(n + 1, k - 1), triangular array read by rows.","first_index":0,"fingerprint":"f6-663a3c2c5c184f3a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["6","3"],["27","27","6"]],"card_rows":[["1"],["6","3"],["27","27","6"],["110","165","75","10"]],"row_sums_preview":["1","9","60","360","2061","11529","63728","350136"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-bb74f03367de3ec8","steps":["basis A007318"],"edges":["e-0c862e94735a3472"],"text":"A145597 \u2192 basis A007318"},"headline":"A145597 \u2192 basis A007318","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,    6;\n  2:  6,   27,    27;\n  3: 10,   75,   165,    110;\n  4: 15,  165,   594,    858,    429;\n  5: 21,  315,  1638,   3822,   4095,    1638;\n  6: 28,  546,  3822,  12740,  21840,   18564,    6188;\n  7: 36,  882,  7938,  35280,  85680,  115668,   81396,   23256;\n  8: 45, 1350, 15120,  85680, 275400,  523260,  581400,  348840,   87210;\n  9: 55, 1980, 26928, 188496, 767448, 1918620, 2984520, 2813976, 1470942, 326876;\n","recurrence_sentence":"The row polynomials satisfy (n^2+6*n+(2*n^2+12*n)*x+(n^2+6*n)*x^2)*P(n,x) = (n^2+8*n+12+(4*n^2+32*n+48)*x+(5*n^2+40*n+60)*x^2+(2*n^2+16*n+24)*x^3)*P(n-1,x) + ((2*n+4)*x+(6*n+12)*x^2+(6*n+12)*x^3+(2*n+4)*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[((n^2 + 8*n + 12 + (4*n^2 + 32*n + 48)*x + (5*n^2 + 40*n + 60)*x^2 + (2*n^2 + 16*n + 24)*x^3)*P[n - 1, x] + ((2*n + 4)*x + (6*n + 12)*x^2 + (6*n + 12)*x^3 + (2*n + 4)*x^4)*D[P[n - 1, x], x])/(n^2 + 6*n + (2*n^2 + 12*n)*x + (n^2 + 6*n)*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] = ((n^2 + 8*n + 12 + (4*n^2 + 32*n + 48)*x + (5*n^2 + 40*n + 60)*x^2 + (2*n^2 + 16*n + 24)*x^3)*v[i - 1] + ((2*n + 4)*x + (6*n + 12)*x^2 + (6*n + 12)*x^3 + (2*n + 4)*x^4)*deriv(v[i - 1], x))/(n^2 + 6*n + (2*n^2 + 12*n)*x + (n^2 + 6*n)*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-bb74f03367de3ec8","steps":["x \u21a6 x/(x+1)"],"edges":["e-8d309dafbf46f610"],"text":"A145597 \u2192 x \u21a6 x/(x+1)"},"headline":"A145597 \u2192 x \u21a6 x/(x+1)","chain":null}
