{"id":"n-9db4cace849774b0","revision":0,"first_index":0,"fingerprint":"f6-9dfcd5dbd8ed9f8f","name":"one_plus_derivative of basis_map_A013610 of Triangle read by rows, s_3(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.","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"],["6","3"],["45","45","9"],["459","621","243","27"],["5994","9720","5265","1134","81"],["95742","175932","116640","35235","4860","243"],["1810836","3653748","2791341","1046115","204120","19683","729"],["39602196","86019084","73131093","31949883","7807590","1071630","76545","2187"],["983520144","2269003752","2099362536","1030601880","296091369","51254532","5235678","288684","6561"],["27348662448","66370603608","65814834420","35448689340","11528136270","2361428559","305873820","24210090","1062882","19683"],["841887574560","2133782381808","2242803605724","1303637763780","468093528045","109163472957","16818513327","1695296790","107173935","3838185","59049"],["28425120994080","74823313013424","82673123861988","51238174718412","19963298615535","5166471528801","910450372293","109615552101","8856464265","457924995","13640319","177147"],["1044517779895680","2842976570338944","3280906398155256","2148532118616312","896837291061426","252972046703376","49677334450029","6867460229058","665041547313","44077716540","1899901575","47829690","531441"],["41499876683848320","116379509654676096","139563809058695112","95889868278782760","42470050681636578","12884964074374392","2768061143719872","427906244182131","47733887130930","3803070980709","210661086636","7689419829","165809592","1594323"],["1772778468170384640","5107155018150730752","6338029782609842256","4543225946216783280","2119039121841216120","684535195902389832","158696516840919057","26922120951037455","3368272425585210","309983001232905","20692664010018","972784150065","30467512530","569173311","4782969"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","9","99","1350","22194","428652","9526572","239660208","6735365136","209203521120","7117993208160","263318896863360","10523998066976640","451935801295937280","20754082122599750400"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["6","3"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-9","18","-9"],["18","-18","0"],["-9","0","0"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-3","6"],["6","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":188499,"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-06ab765377c0e2de"]},{"id":188500,"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-06ab765377c0e2de"]},{"id":188501,"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-06ab765377c0e2de"]},{"id":188502,"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-06ab765377c0e2de"]}],"region_exact":{},"incoming_edges":[{"id":"e-06ab765377c0e2de","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-9db4cace849774b0","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":"bc45e9077ce2bff8180a01065ecc9833ba09d7f86eb04f7392938db51fb959d9","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-5b94261d6f603245","role":"sequence","name":"basis_map_A013610 of Triangle read by rows, s_3(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.","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-03535f54b19fba73","kind":"fingerprint_related","node_a_id":"n-9db4cace849774b0","node_b_id":"n-a08be4d759a7daf2","fingerprint":"f6-9dfcd5dbd8ed9f8f","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-9dfcd5dbd8ed9f8f"},"other_id":"n-a08be4d759a7daf2","other":{"id":"n-a08be4d759a7daf2","name":"one_plus_derivative of basis_map_A013611 of Triangle read by rows, s_4(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.","first_index":0,"fingerprint":"f6-9dfcd5dbd8ed9f8f","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["8","4"],["80","80","16"]],"card_rows":[["1"],["8","4"],["80","80","16"],["1088","1472","576","64"]],"row_sums_preview":["1","12","176","3200","70144","1806336","53526528","1795424256"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-9c5c208df8605708","steps":["basis A013611","D+1"],"edges":["e-8a4919ac2dd1c88f","e-500633f88eec282c"],"text":"A225471 \u2192 basis A013611 \u2192 D+1"},"headline":"A225471 \u2192 basis A013611 \u2192 D+1","chain":null}},{"id":"l-39209e728df4ecff","kind":"fingerprint_related","node_a_id":"n-7c0b893bdd20dbf2","node_b_id":"n-9db4cace849774b0","fingerprint":"f6-9dfcd5dbd8ed9f8f","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-9dfcd5dbd8ed9f8f"},"other_id":"n-7c0b893bdd20dbf2","other":{"id":"n-7c0b893bdd20dbf2","name":"one_plus_derivative of basis_map_A013609 of Triangle of coefficients in expansion of (x+1)*(x+3)*...*(x + 2n - 1) in rising powers of x.","first_index":0,"fingerprint":"f6-9dfcd5dbd8ed9f8f","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["4","2"],["20","20","4"]],"card_rows":[["1"],["4","2"],["20","20","4"],["136","184","72","8"]],"row_sums_preview":["1","6","44","400","4384","56448","836352","14026752"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-4f6e747b794bcc2e","steps":["basis A013609","D+1"],"edges":["e-03658933e548767a","e-c3de98ad74329879"],"text":"A028338 \u2192 basis A013609 \u2192 D+1"},"headline":"A028338 \u2192 basis A013609 \u2192 D+1","chain":null}},{"id":"l-6684368545780280","kind":"fingerprint_related","node_a_id":"n-fb40e29cda26e7b9","node_b_id":"n-9db4cace849774b0","fingerprint":"f6-9dfcd5dbd8ed9f8f","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-9dfcd5dbd8ed9f8f"},"other_id":"n-fb40e29cda26e7b9","other":{"id":"n-fb40e29cda26e7b9","name":"one_plus_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":"f6-9dfcd5dbd8ed9f8f","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","1"],["5","5","1"]],"card_rows":[["1"],["2","1"],["5","5","1"],["17","23","9","1"]],"row_sums_preview":["1","3","11","50","274","1764","13068","109584"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-efe23a76953cf69d","steps":["D+1"],"edges":["e-d9ba82797b30d77a"],"text":"A130534 \u2192 D+1"},"headline":"A130534 \u2192 D+1","chain":null}},{"id":"l-e6ef287923da404f","kind":"fingerprint_related","node_a_id":"n-0254a5a8457ca2c9","node_b_id":"n-9db4cace849774b0","fingerprint":"f6-9dfcd5dbd8ed9f8f","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-9dfcd5dbd8ed9f8f"},"other_id":"n-0254a5a8457ca2c9","other":{"id":"n-0254a5a8457ca2c9","name":"reversal of one_plus_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":"f6-9dfcd5dbd8ed9f8f","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","2"],["1","5","5"]],"card_rows":[["1"],["1","2"],["1","5","5"],["1","9","23","17"]],"row_sums_preview":["1","3","11","50","274","1764","13068","109584"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-efe23a76953cf69d","steps":["D+1","reverse"],"edges":["e-d9ba82797b30d77a","e-46075d03abc6b2c3"],"text":"A130534 \u2192 D+1 \u2192 reverse"},"headline":"A130534 \u2192 D+1 \u2192 reverse","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:           6,           3;\n  2:          45,          45,           9;\n  3:         459,         621,         243,          27;\n  4:        5994,        9720,        5265,        1134,          81;\n  5:       95742,      175932,      116640,       35235,        4860,        243;\n  6:     1810836,     3653748,     2791341,     1046115,      204120,      19683,       729;\n  7:    39602196,    86019084,    73131093,    31949883,     7807590,    1071630,     76545,     2187;\n  8:   983520144,  2269003752,  2099362536,  1030601880,   296091369,   51254532,   5235678,   288684,    6561;\n  9: 27348662448, 66370603608, 65814834420, 35448689340, 11528136270, 2361428559, 305873820, 24210090, 1062882, 19683;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(9*n^2-18*n+9+(18*n-18)*x+9*x^2)*P(n-2,x) + (6*n-3+6*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 6 + 3*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 6 + 3*x;\nP[n_, x_] := P[n, x] = Expand[-(9*n^2 - 18*n + 9 + (18*n - 18)*x + 9*x^2)*P[n - 2, x] + (6*n - 3 + 6*x)*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] = 6 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = -(9*n^2 - 18*n + 9 + (18*n - 18)*x + 9*x^2)*v[i - 2] + (6*n - 3 + 6*x)*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-8229dbdd6a4b5cd9","steps":["basis A013610","D+1"],"edges":["e-86f8b58a2aa07621","e-06ab765377c0e2de"],"text":"A225470 \u2192 basis A013610 \u2192 D+1"},"headline":"A225470 \u2192 basis A013610 \u2192 D+1","chain":null}