{"id":"n-ac1afc976a1e5fa6","revision":0,"first_index":0,"fingerprint":"f6-b8552aefd92cb15e","name":"basis_map_A007318 of Triangle read by rows: T(n,k) = binomial(n-2, k-1) + n*binomial(n-1, k-1), 1 <= k <= n, starting with T(1, 1) = 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":true,"rows":[["1"],["5","2"],["14","13","3"],["36","52","25","4"],["88","172","126","41","5"],["208","512","504","248","61","6"],["480","1424","1760","1160","430","85","7"],["1088","3776","5616","4640","2300","684","113","8"],["2432","9664","16800","16688","10360","4116","1022","145","9"],["5376","24064","47872","55552","41440","20608","6832","1456","181","10"],["11776","58624","131328","174336","151872","90720","37632","10704","1998","221","11"],["25600","140288","349440","522240","520320","362880","180768","64320","16020","2660","265","12"],["55296","330752","906752","1506560","1689600","1347456","783552","334752","104280","23100","3454","313","13"],["118784","770048","2304000","4212736","5251840","4713984","3134208","1562880","584496","161920","32296","4392","365","14"],["253952","1773568","5750784","11474944","15741440","15704832","11751168","6699264","2924064","972400","242528","43992","5486","421","15"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","7","30","117","432","1539","5346","18225","61236","203391","669222","2184813","7085880","22851963","73338858"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A007318","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["5","2"],["14","13","3"]],"recurrence":{"denominator":{"coeffs":[["-2"],["1"]]},"terms":[{"coeff":{"coeffs":[["8","8","2"],["-4","-4","-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-8","-4"],["4","2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":195351,"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-e152509ffbe1cbb9"]},{"id":195352,"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-e152509ffbe1cbb9"]},{"id":195353,"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-e152509ffbe1cbb9"]}],"region_exact":{},"incoming_edges":[{"id":"e-e152509ffbe1cbb9","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-ac1afc976a1e5fa6","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":"f9daef1b794a019b8cb8072d209642e9b0c56ef16b74d1d1440b2e2d302869c2","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-e48eea24800097e1","role":"sequence","name":"Triangle read by rows: T(n,k) = binomial(n-2, k-1) + n*binomial(n-1, k-1), 1 <= k <= n, starting with T(1, 1) = 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:    5,     2;\n  2:   14,    13,     3;\n  3:   36,    52,    25,     4;\n  4:   88,   172,   126,    41,     5;\n  5:  208,   512,   504,   248,    61,     6;\n  6:  480,  1424,  1760,  1160,   430,    85,    7;\n  7: 1088,  3776,  5616,  4640,  2300,   684,  113,    8;\n  8: 2432,  9664, 16800, 16688, 10360,  4116, 1022,  145,   9;\n  9: 5376, 24064, 47872, 55552, 41440, 20608, 6832, 1456, 181, 10;\n","recurrence_sentence":"The row polynomials satisfy (n-2)*P(n,x) = -(4*n-8+(4*n-8)*x+(n-2)*x^2)*P(n-2,x) + (4*n-8+(2*n-4)*x)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 5 + 2*x, P(2,x) = 14 + 13*x + 3*x^2.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 5 + 2*x;\nP[2, x_] := 14 + 13*x + 3*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[(-(4*n - 8 + (4*n - 8)*x + (n - 2)*x^2)*P[n - 2, x] + (4*n - 8 + (2*n - 4)*x)*P[n - 1, x])/(n - 2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 5 + 2*x; v[3] = 14 + 13*x + 3*x^2; for(n = 3, N, my(i = n + 1); v[i] = (-(4*n - 8 + (4*n - 8)*x + (n - 2)*x^2)*v[i - 2] + (4*n - 8 + (2*n - 4)*x)*v[i - 1])/(n - 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-e48eea24800097e1","steps":["basis A007318"],"edges":["e-e152509ffbe1cbb9"],"text":"A123097 \u2192 basis A007318"},"headline":"A123097 \u2192 basis A007318","chain":null}