{"id":"n-b9bba3062ead0647","revision":1,"first_index":0,"fingerprint":"f6-35c9f14f8879aed6","name":"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.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"half_affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[[],["1"],["1"],["2","1"],["6","6"],["24","35","1"],["120","225","15"],["720","1624","175","1"],["5040","13132","1960","28"],["40320","118124","22449","546","1"],["362880","1172700","269325","9450","45"],["3628800","12753576","3416930","157773","1320","1"],["39916800","150917976","45995730","2637558","32670","66"],["479001600","1931559552","657206836","44990231","749463","2717","1"],["6227020800","26596717056","9957703756","790943153","16669653","91091","91"]],"degrees":[-1,0,0,1,1,2,2,3,3,4,4,5,5,6,6],"row_sums":["0","1","1","3","12","60","360","2520","20160","181440","1814400","19958400","239500800","3113510400","43589145600"],"nonn":true,"definitions":[{"position":0,"source":"derived:veronese_section_2_1","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-3","0"],["2","0"],["0","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-4","1"],["4","0"],["-1","0"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":168456,"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-e29866e6e7600a77"]},{"id":168457,"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-e29866e6e7600a77"]},{"id":168458,"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-e29866e6e7600a77"]}],"region_exact":{},"incoming_edges":[{"id":"e-e29866e6e7600a77","operation":{"index_shift":0,"modulus":2,"operation":"veronese_section","residue":1},"operation_detail":"r=2, j=1, shift 0","conclusion_id":"n-b9bba3062ead0647","premise_count":1,"theorem":"Veronese sections of Polya frequency sequences (Aissen--Schoenberg--Whitney--Edrei): if P is real-rooted with coefficients of one sign, equivalently nonzero with all roots in (-inf, 0], every section V_(r,j)(P) is zero or real-rooted with roots in (-inf, 0]","lean_route":null,"lean_declarations":[],"certificate_hash":"d1341d3e020a5edcb12da868d36b4b9e07f6424ffd906c73f08f74d3268db6d4","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-b9fbfb57d9269e2f","role":"sequence","name":"Triangle of unsigned Stirling numbers of the first kind (see A048994), read by rows, T(n,k) for 0 <= k <= n.","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-a97a471d7d88fea0","kind":"fingerprint_related","node_a_id":"n-b9bba3062ead0647","node_b_id":"n-45bd09bce82616a5","fingerprint":"f6-35c9f14f8879aed6","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-35c9f14f8879aed6"},"other_id":"n-45bd09bce82616a5","other":{"id":"n-45bd09bce82616a5","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.","first_index":0,"fingerprint":"f6-35c9f14f8879aed6","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1"],["2","1"]],"card_rows":[["1"],["1"],["2","1"],["6","6"]],"row_sums_preview":["1","1","3","12","60","360","2520","20160"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-efe23a76953cf69d","steps":["Veronese 2"],"edges":["e-4d965342a2d26168"],"text":"A130534 \u2192 Veronese 2"},"headline":"A130534 \u2192 Veronese 2","chain":null}},{"id":"l-6b56b395758982dd","kind":"oeis_match","node_a_id":"n-b9bba3062ead0647","node_b_id":null,"fingerprint":null,"oeis_id":"A001710","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":1,"target":"row_sums","terms_compared":14},"in_catalog":false,"oeis_id":"A001710","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-b2178f004ed9799d","kind":"oeis_match","node_a_id":"n-b9bba3062ead0647","node_b_id":null,"fingerprint":null,"oeis_id":"A188588","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":3,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":2,"target":"row_sums","terms_compared":12},"in_catalog":false,"oeis_id":"A188588","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A001710","relation":"match_row_sums"},{"oeis_id":"A188588","relation":"match_row_sums"}],"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:     0;\n  1:     1;\n  2:     1;\n  3:     2,      1;\n  4:     6,      6;\n  5:    24,     35,     1;\n  6:   120,    225,    15;\n  7:   720,   1624,   175,   1;\n  8:  5040,  13132,  1960,  28;\n  9: 40320, 118124, 22449, 546, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2*n-3)*P(n-1,x) - (n^2-4*n+4-x)*P(n-2,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(2*n - 3)*P[n - 1, x] - (n^2 - 4*n + 4 - x)*P[n - 2, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = (2*n - 3)*v[i - 1] - (n^2 - 4*n + 4 - x)*v[i - 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-b9fbfb57d9269e2f","steps":["Veronese 2 (j=1)"],"edges":["e-e29866e6e7600a77"],"text":"A132393 \u2192 Veronese 2 (j=1)"},"headline":"A132393 \u2192 Veronese 2 (j=1)","chain":null}