{"id":"n-c33cf77401a117b2","revision":0,"first_index":0,"fingerprint":"f6-caf342f7caa2ed5f","name":"derivative of basis_map_A008277 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.","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"],["6","2"],["36","24","3"],["240","240","60","4"],["1800","2400","900","120","5"],["15120","25200","12600","2520","210","6"],["141120","282240","176400","47040","5880","336","7"],["1451520","3386880","2540160","846720","141120","12096","504","8"],["16329600","43545600","38102400","15240960","3175200","362880","22680","720","9"],["199584000","598752000","598752000","279417600","69854400","9979200","831600","39600","990","10"],["2634508800","8781696000","9879408000","5269017600","1536796800","263450880","27442800","1742400","65340","1320","11"],["37362124800","136994457600","171243072000","102745843200","34248614400","6849722880","856215360","67953600","3397680","102960","1716","12"],["566658892800","2266635571200","3116623910400","2077749273600","779155977600","178092794880","25971865920","2473511040","154594440","6246240","156156","2184","13"],["9153720576000","39666122496000","59499183744000","43632734745600","18180306144000","4674935865600","779155977600","86572886400","6492966480","327927600","10930920","229320","2730","14"]],"degrees":[-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["0","1","8","63","544","5225","55656","653023","8379008","116780049","1757211400","28394129951","490371506208","9013522796473","175679564492264"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["1"]],"recurrence":{"denominator":{"coeffs":[["-1"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","0"],["1","1"],["1","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0"],["0","1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":181542,"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-98c76596624efe0c"]},{"id":181543,"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-98c76596624efe0c"]},{"id":181544,"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-98c76596624efe0c"]},{"id":181545,"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-98c76596624efe0c"]}],"region_exact":{},"incoming_edges":[{"id":"e-98c76596624efe0c","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"derivative_polynomial\",\"coefficients\":[{\"numerator\":[\"0\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-c33cf77401a117b2","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":"befab55ddd904860fc65d0817b75cad14475a7c5296668371e9a7e1a510c8b3d","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-011d67b73ad91983","role":"sequence","name":"basis_map_A008277 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.","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-3737b0bd18dc88bc","kind":"fingerprint_related","node_a_id":"n-3a3dbf7079b1a5e6","node_b_id":"n-c33cf77401a117b2","fingerprint":"f6-caf342f7caa2ed5f","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-caf342f7caa2ed5f"},"other_id":"n-3a3dbf7079b1a5e6","other":{"id":"n-3a3dbf7079b1a5e6","name":"basis_map_A008277 of Triangle read by rows: T(n,k) = (-1)^(n+k)*Sum_{j=1..k} s(n,j), where s(n,j) are the signed Stirling numbers of the first kind (n >= 2; 1 <= k <= n-1; s(n,j) = A008275(n,j)).","first_index":0,"fingerprint":"f6-caf342f7caa2ed5f","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["3","1"],["12","8","1"]],"card_rows":[["1"],["3","1"],["12","8","1"],["60","60","15","1"]],"row_sums_preview":["1","4","21","136","1045","9276","93289","1047376"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-914700b1060feeb2","steps":["basis A008277"],"edges":["e-0a932d25d1f1a092"],"text":"A136124 \u2192 basis A008277"},"headline":"A136124 \u2192 basis A008277","chain":null}},{"id":"l-41806807af4a3429","kind":"fingerprint_related","node_a_id":"n-c33cf77401a117b2","node_b_id":"n-5ac2b6c60c089dc4","fingerprint":"f6-caf342f7caa2ed5f","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-caf342f7caa2ed5f"},"other_id":"n-5ac2b6c60c089dc4","other":{"id":"n-5ac2b6c60c089dc4","name":"one_plus_derivative of Triangle read by rows: T(n,k) = binomial(n,k)*(n-1)!/(k-1)!.","first_index":1,"fingerprint":"f6-caf342f7caa2ed5f","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["3","1"],["12","8","1"]],"card_rows":[["1"],["3","1"],["12","8","1"],["60","60","15","1"]],"row_sums_preview":["1","4","21","136","1045","9276","93289","1047376"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-4a0fd05ab0621175","steps":["D+1"],"edges":["e-50c59dba9b7ca874"],"text":"A105278 \u2192 D+1"},"headline":"A105278 \u2192 D+1","chain":null}},{"id":"l-7bbcd9e3981e0a21","kind":"fingerprint_related","node_a_id":"n-271026fda29dd6cb","node_b_id":"n-c33cf77401a117b2","fingerprint":"f6-caf342f7caa2ed5f","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-caf342f7caa2ed5f"},"other_id":"n-271026fda29dd6cb","other":{"id":"n-271026fda29dd6cb","name":"basis_map_A008277 of Unsigned 2-Stirling numbers of the first kind.","first_index":0,"fingerprint":"f6-caf342f7caa2ed5f","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["12","8","1"],["60","60","15","1"],["360","480","180","24","1"]],"card_rows":[["12","8","1"],["60","60","15","1"],["360","480","180","24","1"],["2520","4200","2100","420","35","1"]],"row_sums_preview":["21","136","1045","9276","93289","1047376","12975561","175721140"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c237629cf5fdbb07","steps":["basis A008277"],"edges":["e-13b78624414ee2b1"],"text":"A143491 \u2192 basis A008277"},"headline":"A143491 \u2192 basis A008277","chain":null}},{"id":"l-c1c899129f63c323","kind":"fingerprint_related","node_a_id":"n-c33cf77401a117b2","node_b_id":"n-40d0ed1d7233db60","fingerprint":"f6-caf342f7caa2ed5f","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-caf342f7caa2ed5f"},"other_id":"n-40d0ed1d7233db60","other":{"id":"n-40d0ed1d7233db60","name":"reversal of basis_map_A008277 of Unsigned 2-Stirling numbers of the first kind.","first_index":0,"fingerprint":"f6-caf342f7caa2ed5f","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1","8","12"],["1","15","60","60"],["1","24","180","480","360"]],"card_rows":[["1","8","12"],["1","15","60","60"],["1","24","180","480","360"],["1","35","420","2100","4200","2520"]],"row_sums_preview":["21","136","1045","9276","93289","1047376","12975561","175721140"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c237629cf5fdbb07","steps":["basis A008277","reverse"],"edges":["e-13b78624414ee2b1","e-cb590c605222c451"],"text":"A143491 \u2192 basis A008277 \u2192 reverse"},"headline":"A143491 \u2192 basis A008277 \u2192 reverse","chain":null}},{"id":"l-c9f9871898cc5282","kind":"fingerprint_related","node_a_id":"n-c33cf77401a117b2","node_b_id":"n-b25f7b2d6b8b4be7","fingerprint":"f6-caf342f7caa2ed5f","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-caf342f7caa2ed5f"},"other_id":"n-b25f7b2d6b8b4be7","other":{"id":"n-b25f7b2d6b8b4be7","name":"reversal of basis_map_A008277 of Triangle read by rows: T(n,k) = (-1)^(n+k)*Sum_{j=1..k} s(n,j), where s(n,j) are the signed Stirling numbers of the first kind (n >= 2; 1 <= k <= n-1; s(n,j) = A008275(n,j)).","first_index":0,"fingerprint":"f6-caf342f7caa2ed5f","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","3"],["1","8","12"]],"card_rows":[["1"],["1","3"],["1","8","12"],["1","15","60","60"]],"row_sums_preview":["1","4","21","136","1045","9276","93289","1047376"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-914700b1060feeb2","steps":["basis A008277","reverse"],"edges":["e-0a932d25d1f1a092","e-d28b32295a244135"],"text":"A136124 \u2192 basis A008277 \u2192 reverse"},"headline":"A136124 \u2192 basis A008277 \u2192 reverse","chain":null}},{"id":"l-eed7de0a2d08bf98","kind":"fingerprint_related","node_a_id":"n-c33cf77401a117b2","node_b_id":"n-f6107bd699eaa8cb","fingerprint":"f6-caf342f7caa2ed5f","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-caf342f7caa2ed5f"},"other_id":"n-f6107bd699eaa8cb","other":{"id":"n-f6107bd699eaa8cb","name":"derivative of Triangle read by rows: T(n,k) = binomial(n,k)*(n-1)!/(k-1)!.","first_index":1,"fingerprint":"f6-caf342f7caa2ed5f","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["1"],["6","2"]],"card_rows":[["1"],["6","2"],["36","24","3"],["240","240","60","4"]],"row_sums_preview":["0","1","8","63","544","5225","55656","653023"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-4a0fd05ab0621175","steps":["D"],"edges":["e-2af4f8137776088d"],"text":"A105278 \u2192 D"},"headline":"A105278 \u2192 D","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:        0;\n  1:        1;\n  2:        6,        2;\n  3:       36,       24,        3;\n  4:      240,      240,       60,        4;\n  5:     1800,     2400,      900,      120,       5;\n  6:    15120,    25200,    12600,     2520,     210,      6;\n  7:   141120,   282240,   176400,    47040,    5880,    336,     7;\n  8:  1451520,  3386880,  2540160,   846720,  141120,  12096,   504,   8;\n  9: 16329600, 43545600, 38102400, 15240960, 3175200, 362880, 22680, 720, 9;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = (n^2+n+n*x)*P(n-1,x) + n*x*d/dx P(n-1,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[Together[((n^2 + n + n*x)*P[n - 1, x] + n*x*D[P[n - 1, x], x])/(n - 1)]];\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] = ((n^2 + n + n*x)*v[i - 1] + n*x*deriv(v[i - 1], x))/(n - 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-efe23a76953cf69d","steps":["basis A008277","D"],"edges":["e-31e8b29857f291e5","e-98c76596624efe0c"],"text":"A130534 \u2192 basis A008277 \u2192 D"},"headline":"A130534 \u2192 basis A008277 \u2192 D","chain":null}
