{"id":"n-47cdc739383bdada","revision":1,"first_index":0,"fingerprint":"f6-dd80a37241079ab2","name":"basis_map_A013611 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","12"],["27","108","96"],["110","660","1200","640"],["429","3432","9504","10560","3840"],["1638","16380","61152","104832","80640","21504"],["6188","74256","349440","815360","978432","559104","114688"],["23256","325584","1850688","5483520","9031680","8128512","3612672","589824"],["87210","1395360","9302400","33488640","70502400","87736320","61931520","22118400","2949120"],["326876","5883768","45023616","191009280","491166720","785866752","772079616","441188352","129761280","14417920"],["1225785","24515700","211815648","1035543168","3151653120","6188700672","7858667520","6286934016","2978021376","735313920","69206016"],["4601610","101235420","974859600","5399222400","19005262848","44345613312","69410525184","72114831360","48076554240","19230621696","4048551936","327155712"],["17298645","415167480","4409364960","27296068800","109184275200","295637422080","551856521216","709529812992","616982446080","349000171520","119657201664","21755854848","1526726656"],["65132550","1693446300","19665828000","134601667200","603386784000","1861879219200","4045564723200","6223945728000","6721861386240","4979156582400","2424632770560","721378344960","114504499200","7046430720"],["245642760","6877997280","86704450560","650283379200","3230440012800","11198858711040","27804063006720","49933827440640","64729035571200","59749878988800","38239922552832","16222997446656","4232086290432","591900180480","32212254720"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","18","231","2610","27765","286146","2897468","29045736","289511370","2876724180","28541596941","283039035318","2807268332141","27859382542530","276709333926120"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013611","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["6","12"]],"recurrence":{"denominator":{"coeffs":[["0"],["6"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","0","-32"],["0","0","-48"],["0","0","-16"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["20","40"],["18","36"],["4","8"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":167960,"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-84c19921497cc665"]},{"id":167961,"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-84c19921497cc665"]},{"id":167962,"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-84c19921497cc665"]},{"id":167963,"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-84c19921497cc665"]}],"region_exact":{},"incoming_edges":[{"id":"e-84c19921497cc665","operation":{"map":"A013611","operation":"basis_map"},"operation_detail":"A013611","conclusion_id":"n-47cdc739383bdada","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":"2609849ae393650d58d9d9a51b78c9604146ed30fb251d691940302b6c4432d7","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]"}]}],"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:      6,      12;\n  2:     27,     108,       96;\n  3:    110,     660,     1200,       640;\n  4:    429,    3432,     9504,     10560,      3840;\n  5:   1638,   16380,    61152,    104832,     80640,     21504;\n  6:   6188,   74256,   349440,    815360,    978432,    559104,    114688;\n  7:  23256,  325584,  1850688,   5483520,   9031680,   8128512,   3612672,    589824;\n  8:  87210, 1395360,  9302400,  33488640,  70502400,  87736320,  61931520,  22118400,   2949120;\n  9: 326876, 5883768, 45023616, 191009280, 491166720, 785866752, 772079616, 441188352, 129761280, 14417920;\n","recurrence_sentence":"The row polynomials satisfy (n^2+6*n)*P(n,x) = -(16*n^2+48*n+32)*x^2*P(n-2,x) + (4*n^2+18*n+20+(8*n^2+36*n+40)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 6 + 12*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 6 + 12*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(16*n^2 + 48*n + 32)*x^2*P[n - 2, x] + (4*n^2 + 18*n + 20 + (8*n^2 + 36*n + 40)*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 + 12*x; for(n = 2, N, my(i = n + 1); v[i] = (-(16*n^2 + 48*n + 32)*x^2*v[i - 2] + (4*n^2 + 18*n + 20 + (8*n^2 + 36*n + 40)*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 A013611"],"edges":["e-84c19921497cc665"],"text":"A145597 \u2192 basis A013611"},"headline":"A145597 \u2192 basis A013611","chain":null}