{"id":"n-a2b8ff03da0c14a3","revision":2,"first_index":1,"fingerprint":"f6-322b0642acfc2857","name":"Triangle read by rows: 3-Stirling numbers of the second kind.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["3","1"],["9","7","1"],["27","37","12","1"],["81","175","97","18","1"],["243","781","660","205","25","1"],["729","3367","4081","1890","380","33","1"],["2187","14197","23772","15421","4550","644","42","1"],["6561","58975","133057","116298","47271","9702","1022","52","1"],["19683","242461","724260","830845","447195","124887","18900","1542","63","1"],["59049","989527","3863761","5709330","3961210","1446291","294987","34320","2235","75","1"],["177147","4017157","20308332","38119741","33437800","15531538","4101174","638187","58905","3135","88","1"],["531441","16245775","105558817","249026778","272184341","157690104","52442104","10483044","1286142","96525","4279","102","1"],["1594323","65514541","544039860","1599719485","2154317165","1533705173","629669040","157272544","24630606","2444442","152152","5707","117","1"],["4782969","263652487","2785713841","10142356770","16679939640","14423958549","7200726533","2202394480","428209210","53963910","4422418","232050","7462","133","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","4","17","77","372","1915","10481","60814","372939","2409837","16360786","116393205","865549453","6713065156","54190360453"],"nonn":true,"definitions":[{"position":0,"source":"oeis_catalog:A143495","source_kind":"oeis_catalog","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["3","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":159321,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, 0)","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":false,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0)","region_exact":null,"edges":["e-4d8fac6359540276"]},{"id":159322,"position":1,"claim":"one_signed_coefficients","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":1},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 1: the coefficients of P_n have one sign","region_exact":null,"edges":["e-4d8fac6359540276","e-95a6a1104a7cd304"]},{"id":159323,"position":2,"claim":"hadamard_kernel","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":1},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-4d8fac6359540276","e-95a6a1104a7cd304"]},{"id":159324,"position":3,"claim":"interlacing","from_index":2,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":2,"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 >= 2","region_exact":null,"edges":["e-4d8fac6359540276"]},{"id":159325,"position":4,"claim":"basis_map_preserver","from_index":null,"region_key":null,"provenance":"edges","claim_data":{"claim":"basis_map_preserver","input_interval":"(-inf, 3]","output_interval":"(-inf, 0]","statement":"source (-inf, a + b h + kappa m] = (-inf, 3] with m = 0, target (-inf, 0]"},"label":"Basis map preserves real-rootedness","description":"Basis map u^k -> P_k sends roots in (-inf, 3] to roots in (-inf, 0]: source (-inf, a + b h + kappa m] = (-inf, 3] with m = 0, target (-inf, 0]","region_exact":null,"edges":["e-5a54a7c441e719b3"]},{"id":159326,"position":5,"claim":"real_rooted","from_index":1,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-95a6a1104a7cd304"]},{"id":159327,"position":6,"claim":"interlacing","from_index":2,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2","region_exact":null,"edges":["e-95a6a1104a7cd304"]}],"region_exact":{},"incoming_edges":[{"id":"e-5a54a7c441e719b3","operation":{"operation":"basis_map_theorem"},"operation_detail":null,"conclusion_id":"n-a2b8ff03da0c14a3","premise_count":0,"theorem":"ray basis maps: for L f = (a + b t) f + kappa (t - h) f', b > 0, kappa >= 0, and g real-rooted with every root <= h, of multiplicity m at h, T(u^k) = L^k g sends every real polynomial with all roots in S to one with all roots in (-inf, h] if and only if S lies in (-inf, a + b h + kappa m] (RealRooted rayBasisMap_maps_realRootedIn_iff; the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"6dd591483060725c75418f45908fdb01660c7fd2105510e9a7e3fd161798d949","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"basis_map_preserver","input_interval":"(-inf, 3]","output_interval":"(-inf, 0]","statement":"source (-inf, a + b h + kappa m] = (-inf, 3] with m = 0, target (-inf, 0]"}],"premises":[]},{"id":"e-95a6a1104a7cd304","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"derivative_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-a2b8ff03da0c14a3","premise_count":1,"theorem":"Laguerre's theorem on differential operators: if phi is a real polynomial with only real roots, then phi(D) maps every real-rooted polynomial to a real-rooted polynomial or zero (write phi(D) = C D^j prod (1 + c_i D); D preserves real-rootedness by Rolle's theorem and p + c p' is real-rooted for every real c by Rolle's theorem applied to e^(x/c) p; Polya--Schur 1914); when the coefficients of phi have one sign, coefficients of one sign stay of one sign; applied to phi(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.splits_add_C_mul_derivative_all","RealRooted.splits_C_mul_add_derivative","Polynomial.Splits.derivative"],"certificate_hash":"d9bfa0d70d2f7827149f10318611c1dc8abb09b4fe482e6695d176f4a10143e9","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-6aaec68d653cbdb0","role":"sequence","name":"Triangle read by rows: 2-Stirling numbers of the second kind.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}]},{"id":"e-4d8fac6359540276","operation":{"operation":"theorem_check","theorem":"liu_wang"},"operation_detail":"liu_wang","conclusion_id":"n-a2b8ff03da0c14a3","premise_count":0,"theorem":"L. L. Liu and Y. Wang, A unified approach to polynomial sequences with only real zeros, Adv. in Appl. Math. 38 (2007) 542-560, Theorem 2.1 with g = f' (as in their Corollary 2.4), with nonnegativity of the coefficients replaced by an exact induction keeping every root in (-inf, 0)","lean_route":null,"lean_declarations":[],"certificate_hash":"2452d13598419cb2e46604db19bc6ea8e3a9be75c08435aa5e4b376ed121d107","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":false,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"previous_interlaces_next"}],"premises":[]}],"used_by":[{"id":"e-0e86cfd28f077eaa","operation":"obreschkoff_combination","operation_detail":null,"conclusion_id":"n-88f2be16f47e95a3","conclusion_name":"obreschkoff_combination of Triangle read by rows: 3-Stirling numbers of the second kind."},{"id":"e-0e86cfd28f077eaa","operation":"obreschkoff_combination","operation_detail":null,"conclusion_id":"n-88f2be16f47e95a3","conclusion_name":"obreschkoff_combination of Triangle read by rows: 3-Stirling numbers of the second kind."},{"id":"e-0c475173369de966","operation":"preserver_image","operation_detail":"reversal, shift 0","conclusion_id":"n-06404fc73ac5a6a1","conclusion_name":"reversal of Triangle read by rows: 3-Stirling numbers of the second kind."},{"id":"e-0cb5aa2caeb40953","operation":"preserver_image","operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-2c217d6c9aa4b2f2","conclusion_name":"theta_plus_one_squared of Triangle read by rows: 3-Stirling numbers of the second kind."},{"id":"e-6eaf06ae1521efeb","operation":"preserver_image","operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-8bc7d0e7da25f478","conclusion_name":"derivative of Triangle read by rows: 3-Stirling numbers of the second kind."},{"id":"e-8b885dd48ba326e7","operation":"preserver_image","operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-ab79cd460b94a5bd","conclusion_name":"theta_plus_one of Triangle read by rows: 3-Stirling numbers of the second kind."},{"id":"e-9969ff4d914a2b6f","operation":"preserver_image","operation_detail":"affine, shift 0","conclusion_id":"n-c13c144ce0d4f57d","conclusion_name":"shift_x_plus_one of Triangle read by rows: 3-Stirling numbers of the second kind."},{"id":"e-bbdbb484d8de66ad","operation":"preserver_image","operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-1f64b8c87ab9b182","conclusion_name":"one_plus_derivative of Triangle read by rows: 3-Stirling numbers of the second kind."},{"id":"e-ee2670906a7e6e60","operation":"preserver_image","operation_detail":"mobius, shift 0","conclusion_id":"n-5fdfd6376d0f2ea7","conclusion_name":"mobius_x_over_x_plus_one of Triangle read by rows: 3-Stirling numbers of the second kind."}],"used_by_total":9,"links":[{"id":"l-1f6134d124dfc343","kind":"fingerprint_related","node_a_id":"n-a2b8ff03da0c14a3","node_b_id":"n-4755d98ca7204416","fingerprint":"f6-322b0642acfc2857","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-322b0642acfc2857"},"other_id":"n-4755d98ca7204416","other":{"id":"n-4755d98ca7204416","name":"basis_map_A008277 of Triangle read by rows: T(n,k) is the number of ternary words of length n having k levels (n >= 1, 0 <= k <= n-1). A level is a pair of identical consecutive letters.","first_index":0,"fingerprint":"f6-322b0642acfc2857","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["3"],["9","3"],["27","21","3"]],"card_rows":[["3"],["9","3"],["27","21","3"],["81","111","36","3"]],"row_sums_preview":["3","12","51","231","1116","5745","31443","182442"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-51b4f5e7901c2b50","steps":["basis A008277"],"edges":["e-056f2aef56ba708a"],"text":"A120910 \u2192 basis A008277"},"headline":"A120910 \u2192 basis A008277","chain":null}},{"id":"l-b993a94552b4bc30","kind":"fingerprint_related","node_a_id":"n-a2b8ff03da0c14a3","node_b_id":"n-40f55dba6bf772b4","fingerprint":"f6-322b0642acfc2857","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-322b0642acfc2857"},"other_id":"n-40f55dba6bf772b4","other":{"id":"n-40f55dba6bf772b4","name":"basis_map_A008277 of Triangle whose (i,j)-th entry is binomial(i,j)*2^(i-j).","first_index":0,"fingerprint":"f6-322b0642acfc2857","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["3","1"],["9","7","1"]],"card_rows":[["1"],["3","1"],["9","7","1"],["27","37","12","1"]],"row_sums_preview":["1","4","17","77","372","1915","10481","60814"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-af754f07daf65b5e","steps":["basis A008277"],"edges":["e-74a6e78fdd5274b8"],"text":"A038207 \u2192 basis A008277"},"headline":"A038207 \u2192 basis A008277","chain":null}},{"id":"l-ea3dcdb5fc5e9882","kind":"fingerprint_related","node_a_id":"n-a2b8ff03da0c14a3","node_b_id":"n-06404fc73ac5a6a1","fingerprint":"f6-322b0642acfc2857","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-322b0642acfc2857"},"other_id":"n-06404fc73ac5a6a1","other":{"id":"n-06404fc73ac5a6a1","name":"reversal of Triangle read by rows: 3-Stirling numbers of the second kind.","first_index":1,"fingerprint":"f6-322b0642acfc2857","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","3"],["1","7","9"]],"card_rows":[["1"],["1","3"],["1","7","9"],["1","12","37","27"]],"row_sums_preview":["1","4","17","77","372","1915","10481","60814"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a2b8ff03da0c14a3","steps":["reverse"],"edges":["e-0c475173369de966"],"text":"A143495 \u2192 reverse"},"headline":"A143495 \u2192 reverse","chain":null}},{"id":"l-860eaee23eb8c0b7","kind":"oeis_match","node_a_id":"n-a2b8ff03da0c14a3","node_b_id":null,"fingerprint":null,"oeis_id":"A143495","in_catalog":1,"target":"triangle","row_alignment":"tabl","rows_matched":9,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":3,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":9,"target":"triangle","terms_compared":50},"in_catalog":true,"oeis_id":"A143495","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-d283d649e174579f","kind":"oeis_match","node_a_id":"n-a2b8ff03da0c14a3","node_b_id":null,"fingerprint":null,"oeis_id":"A005494","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":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A005494","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A143495","relation":"identification"},{"oeis_id":"A005494","relation":"match_row_sums"},{"oeis_id":"A143495","relation":"match_triangle"}],"bfiles":[{"node_id":"n-a2b8ff03da0c14a3","oeis_id":"A005494","target":"row_sums","oeis_offset":0,"first_row":1,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":23,"oeis_terms_available":23,"term_count":200,"raw_size":26083},{"node_id":"n-a2b8ff03da0c14a3","oeis_id":"A143495","target":"triangle","oeis_offset":3,"first_row":1,"rows_used":44,"terms_matched":50,"oeis_terms_agreeing":50,"oeis_terms_available":50,"term_count":990,"raw_size":23365}],"oeis_texts":[{"oeis_id":"","target":"node","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:     1;\n   2:     3,      1;\n   3:     9,      7,      1;\n   4:    27,     37,     12,      1;\n   5:    81,    175,     97,     18,      1;\n   6:   243,    781,    660,    205,     25,      1;\n   7:   729,   3367,   4081,   1890,    380,     33,     1;\n   8:  2187,  14197,  23772,  15421,   4550,    644,    42,    1;\n   9:  6561,  58975, 133057, 116298,  47271,   9702,  1022,   52,  1;\n  10: 19683, 242461, 724260, 830845, 447195, 124887, 18900, 1542, 63, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (3+x)*P(n-1,x) + x*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(3 + x)*P[n - 1, x] + x*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (3 + x)*v[i - 1] + x*deriv(v[i - 1], x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}},{"oeis_id":"A143495","target":"triangle","first_row":1,"oeis_offset":3,"reversed":false,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   3:     1;\n   4:     3,      1;\n   5:     9,      7,      1;\n   6:    27,     37,     12,      1;\n   7:    81,    175,     97,     18,      1;\n   8:   243,    781,    660,    205,     25,      1;\n   9:   729,   3367,   4081,   1890,    380,     33,     1;\n  10:  2187,  14197,  23772,  15421,   4550,    644,    42,    1;\n  11:  6561,  58975, 133057, 116298,  47271,   9702,  1022,   52,  1;\n  12: 19683, 242461, 724260, 830845, 447195, 124887, 18900, 1542, 63, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (3+x)*P(n-1,x) + x*d/dx P(n-1,x) for n >= 4, with P(3,x) = 1.","mathematica":"P[3, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(3 + x)*P[n - 1, x] + x*D[P[n - 1, x], x]];\nTable[PadRight[CoefficientList[P[n, x], x], n - 2], {n, 3, 12}] // Flatten","pari":"P(N) = {my(v = vector(N - 2)); v[1] = 1; for(n = 4, N, my(i = n - 2); v[i] = (3 + x)*v[i - 1] + x*deriv(v[i - 1], x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - 2 - #c)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i + 2, v[i])))};\nT(12)","reading":{"first_row":1,"offset":3,"reversed":false,"trim_zeros":"none","regular_triangle":true}}],"derivation":{"root":"n-a2b8ff03da0c14a3","steps":[],"edges":[],"text":"A143495"},"headline":"A143495: Triangle read by rows: 3-Stirling numbers of the second kind.","chain":null}