{"id":"n-714479c216bddb04","revision":0,"first_index":0,"fingerprint":"f6-792365468f8ca6c1","name":"Triangle of coefficients, read by rows, where the n-th row forms the polynomial P(n,x) = {Sum_{k=1..n} 1/(k+x)}*{Product_{k=1..n} (k+x)}.","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"],["3","2"],["11","12","3"],["50","70","30","4"],["274","450","255","60","5"],["1764","3248","2205","700","105","6"],["13068","26264","20307","7840","1610","168","7"],["109584","236248","201852","89796","22680","3276","252","8"],["1026576","2345400","2171040","1077300","316365","56700","6090","360","9"],["10628640","25507152","25228500","13667720","4510275","946638","127050","10560","495","10"],["120543840","301835952","315774228","183982920","66697675","15825348","2501961","261360","17325","660","11"],["1486442880","3863119104","4242044664","2628827344","1030350750","269941386","48486438","5995704","501930","27170","858","12"],["19802759040","53193434112","60941259288","39830815024","16680593930","4745658918","945255311","133357224","13270257","910910","41041","1092","13"],["283465647360","784313595648","932967781200","638886422720","283316833800","86455937568","18770176425","2947292920","335810475","27497470","1576575","60060","1365","14"],["4339163001600","12331635229440","15170987111472","10827253382400","5048360535400","1636819264080","382417906871","65661024000","8352861660","785584800","53895842","2620800","85540","1680","15"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","5","26","154","1044","8028","69264","663696","6999840","80627040","1007441280","13575738240","196287356160","3031488633600","49811492505600"],"nonn":true,"definitions":[{"position":0,"source":"oeis_catalog:A074246","source_kind":"oeis_catalog","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["3","2"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","2"],["2","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","-1"],["0","-2","0"],["-1","0","0"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":169183,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, +inf)","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)","region_exact":null,"edges":["e-8d7bc643ffd1dd86"]},{"id":169184,"position":1,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"previous_interlaces_next"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m >= 1","region_exact":null,"edges":["e-8d7bc643ffd1dd86"]}],"region_exact":{},"incoming_edges":[{"id":"e-8d7bc643ffd1dd86","operation":{"operation":"theorem_check","theorem":"liu_wang_three_term"},"operation_detail":"liu_wang_three_term","conclusion_id":"n-714479c216bddb04","premise_count":0,"theorem":"L. L. Liu and Y. Wang, Adv. Appl. Math. 38 (2007) 542-560, Theorem 2.1 and Corollary 2.4 (A unified approach to polynomial sequences with only real zeros; the lag row as the interlacer, b_n = 0), weak interlacing of consecutive rows, with nonnegativity of the coefficients replaced, where a root region is certified, by an exact root-location induction; Proposition 3.3 (the case B_n(x) = -(c_n x + d_n)^2)","lean_route":"sign","lean_declarations":["RealRooted.threeTerm_interlaces_of_eval_nonpos","RealRooted.threeTerm_interlaces","RealRooted.eval_nonpos_of_natDegree_le_two"],"certificate_hash":"df0aa5730c9fc4b48c728f137583c04b4b1332a6f3185d037564e4170d4f2771","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"previous_interlaces_next"}],"premises":[]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-0e012f647797e93b","kind":"fingerprint_related","node_a_id":"n-a34e8f08502aec80","node_b_id":"n-714479c216bddb04","fingerprint":"f6-792365468f8ca6c1","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-792365468f8ca6c1"},"other_id":"n-a34e8f08502aec80","other":{"id":"n-a34e8f08502aec80","name":"derivative 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":"f6-792365468f8ca6c1","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["1"],["3","2"]],"card_rows":[["1"],["3","2"],["11","12","3"],["50","70","30","4"]],"row_sums_preview":["0","1","5","26","154","1044","8028","69264"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-efe23a76953cf69d","steps":["D"],"edges":["e-9a8da9628cbb79fd"],"text":"A130534 \u2192 D"},"headline":"A130534 \u2192 D","chain":null}},{"id":"l-00ac3b6832fdb1c3","kind":"oeis_match","node_a_id":"n-714479c216bddb04","node_b_id":null,"fingerprint":null,"oeis_id":"A001705","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":1,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A001705","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-123855019ca79f52","kind":"oeis_match","node_a_id":"n-714479c216bddb04","node_b_id":null,"fingerprint":null,"oeis_id":"A074246","in_catalog":1,"target":"triangle","row_alignment":"tabl","rows_matched":9,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":1,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":9,"target":"triangle","terms_compared":45},"in_catalog":true,"oeis_id":"A074246","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A074246","relation":"identification"},{"oeis_id":"A001705","relation":"match_row_sums"},{"oeis_id":"A074246","relation":"match_triangle"}],"bfiles":[{"node_id":"n-714479c216bddb04","oeis_id":"A074246","target":"triangle","oeis_offset":1,"first_row":0,"rows_used":44,"terms_matched":45,"oeis_terms_agreeing":45,"oeis_terms_available":45,"term_count":990,"raw_size":28514}],"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:        3,        2;\n  2:       11,       12,        3;\n  3:       50,       70,       30,        4;\n  4:      274,      450,      255,       60,       5;\n  5:     1764,     3248,     2205,      700,     105,      6;\n  6:    13068,    26264,    20307,     7840,    1610,    168,      7;\n  7:   109584,   236248,   201852,    89796,   22680,   3276,    252,     8;\n  8:  1026576,  2345400,  2171040,  1077300,  316365,  56700,   6090,   360,   9;\n  9: 10628640, 25507152, 25228500, 13667720, 4510275, 946638, 127050, 10560, 495, 10;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2*n+1+2*x)*P(n-1,x) - (n^2+2*n*x+x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 2*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3 + 2*x;\nP[n_, x_] := P[n, x] = Expand[(2*n + 1 + 2*x)*P[n - 1, x] - (n^2 + 2*n*x + x^2)*P[n - 2, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (2*n + 1 + 2*x)*v[i - 1] - (n^2 + 2*n*x + x^2)*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}},{"oeis_id":"A074246","target":"triangle","first_row":0,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:        1;\n   2:        3,        2;\n   3:       11,       12,        3;\n   4:       50,       70,       30,        4;\n   5:      274,      450,      255,       60,       5;\n   6:     1764,     3248,     2205,      700,     105,      6;\n   7:    13068,    26264,    20307,     7840,    1610,    168,      7;\n   8:   109584,   236248,   201852,    89796,   22680,   3276,    252,     8;\n   9:  1026576,  2345400,  2171040,  1077300,  316365,  56700,   6090,   360,   9;\n  10: 10628640, 25507152, 25228500, 13667720, 4510275, 946638, 127050, 10560, 495, 10;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2*n-1+2*x)*P(n-1,x) - (n^2-2*n+1+(2*n-2)*x+x^2)*P(n-2,x) for n >= 3, with P(1,x) = 1, P(2,x) = 3 + 2*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 3 + 2*x;\nP[n_, x_] := P[n, x] = Expand[(2*n - 1 + 2*x)*P[n - 1, x] - (n^2 - 2*n + 1 + (2*n - 2)*x + x^2)*P[n - 2, x]];\nTable[PadRight[CoefficientList[P[n, x], x], n], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 3 + 2*x; for(n = 3, N, my(i = n); v[i] = (2*n - 1 + 2*x)*v[i - 1] - (n^2 - 2*n + 1 + (2*n - 2)*x + x^2)*v[i - 2]); v};\nrow(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":0,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":true}}],"derivation":{"root":"n-714479c216bddb04","steps":[],"edges":[],"text":"A074246"},"headline":"A074246: Triangle of coefficients, read by rows, where the n-th row forms the polynomial P(n,x) = {Sum_{k=1..n} 1/(k+x)}*{Product_{k=1..n} (k+x)}.","chain":null}
