{"id":"n-4355c823139108ab","revision":2,"first_index":0,"fingerprint":"f6-663a3c2c5c184f3a","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.","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"],["6","3"],["27","27","6"],["110","165","75","10"],["429","858","594","165","15"],["1638","4095","3822","1638","315","21"],["6188","18564","21840","12740","3822","546","28"],["23256","81396","115668","85680","35280","7938","882","36"],["87210","348840","581400","523260","275400","85680","15120","1350","45"],["326876","1470942","2813976","2984520","1918620","767448","188496","26928","1980","55"],["1225785","6128925","13238478","16180362","12311145","6043653","1918620","383724","45441","2805","66"],["4601610","25308855","60928725","84362850","74239308","43306263","16945929","4401540","733590","73359","3861","78"],["17298645","103791870","275585310","426501075","426501075","288708420","134730596","43306263","9414405","1331330","114114","5187","91"],["65132550","423361575","1229114250","2103151050","2356979625","1818241425","987686700","379879500","102567465","18993975","2312310","171990","6825","105"],["245642760","1719499320","5419028160","10160677800","12618906300","10936385460","6788101320","3047718960","987686700","227927700","36468432","3867864","252252","8820","120"]],"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:basis_map_A007318","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["6","3"]],"recurrence":{"denominator":{"coeffs":[["0"],["6"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","0","-2"],["0","0","-3"],["0","0","-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["20","10"],["18","9"],["4","2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}},{"position":1,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":"e-05e5d91dd5b215be","recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["0"],["6"],["1"]]},"terms":[{"coeff":{"coeffs":[["20","8"],["18","10"],["4","3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-4","-4"],["0","-2","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":161750,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-0c862e94735a3472"]},{"id":161751,"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-0c862e94735a3472","e-9e81eec4f5e94153"]},{"id":161752,"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-0c862e94735a3472","e-9e81eec4f5e94153"]},{"id":161753,"position":3,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1","region_exact":null,"edges":["e-0c862e94735a3472","e-9e81eec4f5e94153"]},{"id":161754,"position":4,"claim":"real_rooted","from_index":0,"region_key":"(-inf, -1]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, -1]","region_exact":null,"edges":["e-9e81eec4f5e94153"]}],"region_exact":{},"incoming_edges":[{"id":"e-0c862e94735a3472","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-4355c823139108ab","premise_count":1,"theorem":"geometric family P_k = g(t) (a + d t)^k: T(p) = g(t) p(a + d t), so the roots of T(p) are those of g and the points (r - a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"12cc83e917ef266d895b6bfcb61f749b2a8c225804d9215b9059ca0a20ccfd45","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"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]"}]},{"id":"e-05e5d91dd5b215be","operation":{"operation":"definition_equality"},"operation_detail":null,"conclusion_id":"n-4355c823139108ab","premise_count":0,"theorem":"all-index identity Q_n = P_n certified by exact Ore-algebra division and identifying rows (polytool preserver-image certificate of the identity map)","lean_route":null,"lean_declarations":[],"certificate_hash":"162043e20d91fea4ba2676112a3480315fbbdaae3d4330705af0721cd8042b10","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[],"premises":[]},{"id":"e-9e81eec4f5e94153","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-4355c823139108ab","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":null,"lean_declarations":["Polynomial.Splits.comp_of_degree_le_one"],"certificate_hash":"758fb261960caba8b39ddbe96c9eed09fabd46e3ccc1f7ac0e62fc10d84838a5","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"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-1a7a27d6b2026223","other":{"id":"n-1a7a27d6b2026223","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.","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"],["3","6"],["6","27","27"]],"card_rows":[["1"],["3","6"],["6","27","27"],["10","75","165","110"]],"row_sums_preview":["1","9","60","360","2061","11529","63728","350136"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"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}}],"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:      6,       3;\n  2:     27,      27,       6;\n  3:    110,     165,      75,      10;\n  4:    429,     858,     594,     165,      15;\n  5:   1638,    4095,    3822,    1638,     315,     21;\n  6:   6188,   18564,   21840,   12740,    3822,    546,     28;\n  7:  23256,   81396,  115668,   85680,   35280,   7938,    882,    36;\n  8:  87210,  348840,  581400,  523260,  275400,  85680,  15120,  1350,   45;\n  9: 326876, 1470942, 2813976, 2984520, 1918620, 767448, 188496, 26928, 1980, 55;\n","recurrence_sentence":"The row polynomials satisfy (n^2+6*n)*P(n,x) = -(n^2+3*n+2)*x^2*P(n-2,x) + (4*n^2+18*n+20+(2*n^2+9*n+10)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 6 + 3*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 6 + 3*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^2 + 3*n + 2)*x^2*P[n - 2, x] + (4*n^2 + 18*n + 20 + (2*n^2 + 9*n + 10)*x)*P[n - 1, x])/(n^2 + 6*n)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 6 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^2 + 3*n + 2)*x^2*v[i - 2] + (4*n^2 + 18*n + 20 + (2*n^2 + 9*n + 10)*x)*v[i - 1])/(n^2 + 6*n)); 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":["basis A007318"],"edges":["e-0c862e94735a3472"],"text":"A145597 \u2192 basis A007318"},"headline":"A145597 \u2192 basis A007318","chain":null}
