{"id":"n-3a7332e45feb5b75","revision":0,"first_index":0,"fingerprint":"f6-ef4177b197ad2251","name":"basis_map_A013610 of veronese_section_2_0 of Triangle T(n,k), 0 <= k <= n, read by rows, giving coefficients of the polynomial (x+1)(x+2)...(x+n), expanded in increasing powers of x. T(n,k) is also the unsigned Stirling number |s(n+1, k+1)|, denoting the number of permutations on n+1 elements that contain exactly k+1 cycles.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"half_affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["1"],["3","3"],["12","18"],["60","111","9"],["360","765","135"],["2520","5931","1602","27"],["20160","51408","18396","756"],["181440","493992","216837","14850","81"],["1814400","5219640","2681505","260010","3645"],["19958400","60198120","35083611","4402701","107325","243"],["239500800","752859360","486945756","74760246","2673000","16038"],["3113510400","10151866080","7170313428","1296412371","61808103","661689","729"],["43589145600","146856265920","111883170672","23180431365","1387244313","22267791","66339"],["653837184000","2268946209600","1846688550480","429585308811","30961072260","675490914","3653748","2187"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","1","6","30","180","1260","10080","90720","907200","9979200","119750400","1556755200","21794572800","326918592000","5230697472000"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013610","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","3"],["2","0"],["-1","0"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-1"],["2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":202441,"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-b1ce9a4ede2a6652"]},{"id":202442,"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-b1ce9a4ede2a6652"]},{"id":202443,"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-b1ce9a4ede2a6652"]},{"id":202444,"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-b1ce9a4ede2a6652"]}],"region_exact":{},"incoming_edges":[{"id":"e-b1ce9a4ede2a6652","operation":{"map":"A013610","operation":"basis_map"},"operation_detail":"A013610","conclusion_id":"n-3a7332e45feb5b75","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":"2de4d930402365c13718db1288ddc98a9c8a729be8037701ef60b7cc9e7b8c73","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-45bd09bce82616a5","role":"sequence","name":"veronese_section_2_0 of Triangle T(n,k), 0 <= k <= n, read by rows, giving coefficients of the polynomial (x+1)(x+2)...(x+n), expanded in increasing powers of x. T(n,k) is also the unsigned Stirling number |s(n+1, k+1)|, denoting the number of permutations on n+1 elements that contain exactly k+1 cycles.","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-31c7bcafdb1669b3","kind":"fingerprint_related","node_a_id":"n-b87ce51f38cbd48e","node_b_id":"n-3a7332e45feb5b75","fingerprint":"f6-ef4177b197ad2251","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-ef4177b197ad2251"},"other_id":"n-b87ce51f38cbd48e","other":{"id":"n-b87ce51f38cbd48e","name":"basis_map_A013610 of veronese_section_2_1 of Triangle of unsigned Stirling numbers of the first kind (see A048994), read by rows, T(n,k) for 0 <= k <= n.","first_index":0,"fingerprint":"f6-ef4177b197ad2251","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["1"],["1"]],"card_rows":[["1"],["1"],["3","3"],["12","18"]],"row_sums_preview":["0","1","1","6","30","180","1260","10080"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-b9fbfb57d9269e2f","steps":["Veronese 2 (j=1)","basis A013610"],"edges":["e-b4f12c588dbcf356","e-4ba9ab5145d28bd0"],"text":"A132393 \u2192 Veronese 2 (j=1) \u2192 basis A013610"},"headline":"A132393 \u2192 Veronese 2 (j=1) \u2192 basis A013610","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:       1;\n  2:       3,       3;\n  3:      12,      18;\n  4:      60,     111,       9;\n  5:     360,     765,     135;\n  6:    2520,    5931,    1602,     27;\n  7:   20160,   51408,   18396,    756;\n  8:  181440,  493992,  216837,  14850,   81;\n  9: 1814400, 5219640, 2681505, 260010, 3645;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n^2-2*n-3*x)*P(n-2,x) + (2*n-1)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[-(n^2 - 2*n - 3*x)*P[n - 2, x] + (2*n - 1)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = -(n^2 - 2*n - 3*x)*v[i - 2] + (2*n - 1)*v[i - 1]); 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-efe23a76953cf69d","steps":["Veronese 2","basis A013610"],"edges":["e-4d965342a2d26168","e-b1ce9a4ede2a6652"],"text":"A130534 \u2192 Veronese 2 \u2192 basis A013610"},"headline":"A130534 \u2192 Veronese 2 \u2192 basis A013610","chain":null}