{"id":"n-fd92598b7a526ebf","revision":0,"first_index":0,"fingerprint":"f6-adb63b75c11d63d7","name":"Triangle T(n,k) read by rows: associated Stirling numbers of first kind (n >= 0 and 0 <= k <= floor(n/2)).","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"],[],["0","1"],["0","2"],["0","6","3"],["0","24","20"],["0","120","130","15"],["0","720","924","210"],["0","5040","7308","2380","105"],["0","40320","64224","26432","2520"],["0","362880","623376","303660","44100","945"],["0","3628800","6636960","3678840","705320","34650"],["0","39916800","76998240","47324376","11098780","866250","10395"],["0","479001600","967524480","647536032","177331440","18858840","540540"],["0","6227020800","13096736640","9418945536","2920525608","389449060","18288270","135135"]],"degrees":[0,-1,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","0","1","2","9","44","265","1854","14833","133496","1334961","14684570","176214841","2290792932","32071101049"],"nonn":true,"definitions":[{"position":0,"source":"oeis_catalog:A106828","source_kind":"oeis_catalog","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["0"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-1"],["1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-1"],["0","1"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":169315,"position":0,"claim":"real_rooted","from_index":2,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":2,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 2, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-8ab2f783ccefdea0"]},{"id":169316,"position":1,"claim":"one_signed_coefficients","from_index":2,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":2},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 2: the coefficients of P_n have one sign","region_exact":null,"edges":["e-8ab2f783ccefdea0"]},{"id":169317,"position":2,"claim":"hadamard_kernel","from_index":2,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":2},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 2, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-8ab2f783ccefdea0"]},{"id":169318,"position":3,"claim":"interlacing","from_index":3,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":3,"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 >= 3","region_exact":null,"edges":["e-8ab2f783ccefdea0"]}],"region_exact":{},"incoming_edges":[{"id":"e-8ab2f783ccefdea0","operation":{"operation":"theorem_check","theorem":"liu_wang_three_term"},"operation_detail":"liu_wang_three_term","conclusion_id":"n-fd92598b7a526ebf","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","lean_route":null,"lean_declarations":["RealRooted.MaWang.strictInterl_of_interlaces_evalCoeff_nonpos"],"certificate_hash":"cffa42de753197bdc757fbfb2b788d5f400d19140a6cd51257211a7b22116a2f","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":2,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":3,"kind":"weak","orientation":"previous_interlaces_next"}],"premises":[]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-a6092d7b4639690e","kind":"fingerprint_related","node_a_id":"n-fd92598b7a526ebf","node_b_id":"n-148a066ece673d8a","fingerprint":"f6-adb63b75c11d63d7","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-adb63b75c11d63d7"},"other_id":"n-148a066ece673d8a","other":{"id":"n-148a066ece673d8a","name":"Regular triangle read by rows. T(n, k) = [[n, k]], where [[n, k]] are the second order Stirling cycle numbers (or second order reciprocal Stirling numbers). T(n, k) for 0 <= k <= n.","first_index":1,"fingerprint":"f6-adb63b75c11d63d7","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],[],["0","1"]],"card_rows":[["1"],["0","1"],["0","2"],["0","6","3"]],"row_sums_preview":["1","0","1","2","9","44","265","1854"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":["A358622"],"derivation":{"root":"n-148a066ece673d8a","steps":[],"edges":[],"text":"A358622"},"headline":"A358622: Regular triangle read by rows. T(n, k) = [[n, k]], where [[n, k]] are the second order Stirling cycle numbers (or second order reciprocal Stirling numbers). T(n, k) for 0 <= k <= n.","chain":null}},{"id":"l-b3ee0ec27c68beb6","kind":"fingerprint_related","node_a_id":"n-f13c6b611f4ce3bb","node_b_id":"n-fd92598b7a526ebf","fingerprint":"f6-adb63b75c11d63d7","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-adb63b75c11d63d7"},"other_id":"n-f13c6b611f4ce3bb","other":{"id":"n-f13c6b611f4ce3bb","name":"Triangle T(n,k) read by rows: associated Stirling numbers of first kind (n >= 2, 1 <= k <= floor(n/2)).","first_index":0,"fingerprint":"f6-adb63b75c11d63d7","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2"],["6","3"]],"card_rows":[["1"],["2"],["6","3"],["24","20"]],"row_sums_preview":["1","2","9","44","265","1854","14833","133496"],"regions":["(-inf, 0)","(-inf, 0]"],"region_exact":{},"oeis_ids":["A008306"],"derivation":{"root":"n-f13c6b611f4ce3bb","steps":[],"edges":[],"text":"A008306"},"headline":"A008306: Triangle T(n,k) read by rows: associated Stirling numbers of first kind (n >= 2, 1 <= k <= floor(n/2)).","chain":null}},{"id":"l-c411ab942529128a","kind":"fingerprint_related","node_a_id":"n-f364085508d20047","node_b_id":"n-fd92598b7a526ebf","fingerprint":"f6-adb63b75c11d63d7","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-adb63b75c11d63d7"},"other_id":"n-f364085508d20047","other":{"id":"n-f364085508d20047","name":"Triangle T(n,k) read by rows; related to number of preorders.","first_index":0,"fingerprint":"f6-adb63b75c11d63d7","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["0","2"],["0","3","6"]],"card_rows":[["1"],["0","2"],["0","3","6"],["0","0","20","24"]],"row_sums_preview":["1","2","9","44","265","1854","14833","133496"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":["A079510"],"derivation":{"root":"n-f364085508d20047","steps":[],"edges":[],"text":"A079510"},"headline":"A079510: Triangle T(n,k) read by rows; related to number of preorders.","chain":null}},{"id":"l-e569061e6a7e4f22","kind":"fingerprint_related","node_a_id":"n-fd92598b7a526ebf","node_b_id":"n-5f40aeb2fcfc43d2","fingerprint":"f6-adb63b75c11d63d7","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-adb63b75c11d63d7"},"other_id":"n-5f40aeb2fcfc43d2","other":{"id":"n-5f40aeb2fcfc43d2","name":"shift_x_plus_one of Triangle read by rows: T(n,k) is the number of permutations of an n-set having k cycles of size > 1 (0<=k<=floor(n/2)).","first_index":0,"fingerprint":"f6-adb63b75c11d63d7","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","5"]],"row_sums_preview":["1","1","3","11","53","309","2119","16687"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-497753d6192915db","steps":["basis A007318"],"edges":["e-9e748e197fdf43a2"],"text":"A136394 \u2192 basis A007318"},"headline":"A136394 \u2192 basis A007318","chain":null}},{"id":"l-2b8f64a105dba7c9","kind":"oeis_match","node_a_id":"n-fd92598b7a526ebf","node_b_id":null,"fingerprint":null,"oeis_id":"A257953","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":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A257953","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-2e02198f59a95b7d","kind":"oeis_match","node_a_id":"n-fd92598b7a526ebf","node_b_id":null,"fingerprint":null,"oeis_id":"A008306","in_catalog":1,"target":"triangle","row_alignment":"node_lengths","rows_matched":11,"oeis_nonn":1,"details":{"alignment":{"first_row":2,"low_zeros_dropped":true,"oeis_offset":2,"oeis_term_position":0,"row_alignment":"node_lengths","rows_matched":11,"target":"triangle","terms_compared":36},"in_catalog":true,"oeis_id":"A008306","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-b6b55eef99a25549","kind":"oeis_match","node_a_id":"n-fd92598b7a526ebf","node_b_id":null,"fingerprint":null,"oeis_id":"A260216","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":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A260216","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-b7d8c8148f12eeb3","kind":"oeis_match","node_a_id":"n-fd92598b7a526ebf","node_b_id":null,"fingerprint":null,"oeis_id":"A000166","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":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A000166","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-c63282df7cf958dc","kind":"oeis_match","node_a_id":"n-fd92598b7a526ebf","node_b_id":null,"fingerprint":null,"oeis_id":"A260091","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":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A260091","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-fbf4f4da11fc7c39","kind":"oeis_match","node_a_id":"n-fd92598b7a526ebf","node_b_id":null,"fingerprint":null,"oeis_id":"A260115","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":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A260115","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A106828","relation":"identification"},{"oeis_id":"A000166","relation":"match_row_sums"},{"oeis_id":"A257953","relation":"match_row_sums"},{"oeis_id":"A260091","relation":"match_row_sums"},{"oeis_id":"A260115","relation":"match_row_sums"},{"oeis_id":"A260216","relation":"match_row_sums"},{"oeis_id":"A008306","relation":"match_triangle"}],"bfiles":[{"node_id":"n-fd92598b7a526ebf","oeis_id":"A000166","target":"row_sums","oeis_offset":0,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":24,"oeis_terms_available":24,"term_count":200,"raw_size":33972},{"node_id":"n-fd92598b7a526ebf","oeis_id":"A008306","target":"triangle","oeis_offset":2,"first_row":2,"rows_used":62,"terms_matched":36,"oeis_terms_agreeing":36,"oeis_terms_available":36,"term_count":992,"raw_size":49572}],"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: 0;\n  2: 0,     1;\n  3: 0,     2;\n  4: 0,     6,     3;\n  5: 0,    24,    20;\n  6: 0,   120,   130,    15;\n  7: 0,   720,   924,   210;\n  8: 0,  5040,  7308,  2380,  105;\n  9: 0, 40320, 64224, 26432, 2520;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (n-1)*P(n-1,x) + (n-1)*x*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 0.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 0;\nP[n_, x_] := P[n, x] = Expand[(n - 1)*P[n - 1, x] + (n - 1)*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] = 1; v[2] = 0; for(n = 2, N, my(i = n + 1); v[i] = (n - 1)*v[i - 1] + (n - 1)*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}},{"oeis_id":"A008306","target":"triangle","first_row":2,"oeis_offset":2,"reversed":false,"trim_zeros":"low","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   2:       1;\n   3:       2;\n   4:       6,       3;\n   5:      24,      20;\n   6:     120,     130,      15;\n   7:     720,     924,     210;\n   8:    5040,    7308,    2380,    105;\n   9:   40320,   64224,   26432,   2520;\n  10:  362880,  623376,  303660,  44100,   945;\n  11: 3628800, 6636960, 3678840, 705320, 34650;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in ascending powers of x, starting at the lowest nonzero coefficient, where P(n,x) = (n-1)*P(n-1,x) + (n-1)*x*P(n-2,x) for n >= 4, with P(2,x) = x, P(3,x) = 2*x.","mathematica":"P[2, x_] := x;\nP[3, x_] := 2*x;\nP[n_, x_] := P[n, x] = Expand[(n - 1)*P[n - 1, x] + (n - 1)*x*P[n - 2, x]];\nTable[CoefficientList[Expand[P[n, x]/x^Exponent[P[n, x], x, Min]], x], {n, 2, 11}] // Flatten","pari":"P(N) = {my(v = vector(N - 1)); v[1] = x; v[2] = 2*x; for(n = 4, N, my(i = n - 1); v[i] = (n - 1)*v[i - 1] + (n - 1)*x*v[i - 2]); v};\nrow(n, p) = {my(c = Vecrev(p/x^valuation(p, x))); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i + 1, v[i])))};\nT(11)","reading":{"first_row":2,"offset":2,"reversed":false,"trim_zeros":"low","regular_triangle":false}}],"derivation":{"root":"n-fd92598b7a526ebf","steps":[],"edges":[],"text":"A106828"},"headline":"A106828: Triangle T(n,k) read by rows: associated Stirling numbers of first kind (n >= 0 and 0 <= k <= floor(n/2)).","chain":null}
