{"id":"n-0c345ddac20c06a3","revision":0,"first_index":1,"fingerprint":"f6-f3c222c4860e5244","name":"one_plus_derivative of Triangle read by rows: the unsigned Lah numbers T(n, k) = binomial(n-1, k-1)*n!/k! if n > 0 and k > 0, T(n, 0) = 0^n and otherwise 0, for n >= 0 and 0 <= k <= n.","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"],["1","1"],["2","4","1"],["6","18","9","1"],["24","96","72","16","1"],["120","600","600","200","25","1"],["720","4320","5400","2400","450","36","1"],["5040","35280","52920","29400","7350","882","49","1"],["40320","322560","564480","376320","117600","18816","1568","64","1"],["362880","3265920","6531840","5080320","1905120","381024","42336","2592","81","1"],["3628800","36288000","81648000","72576000","31752000","7620480","1058400","86400","4050","100","1"],["39916800","439084800","1097712000","1097712000","548856000","153679680","25613280","2613600","163350","6050","121","1"],["479001600","5748019200","15807052800","17563392000","9879408000","3161410560","614718720","75271680","5880600","290400","8712","144","1"],["6227020800","80951270400","242853811200","296821324800","185513328000","66784798080","14841066240","2120152320","198764280","12269400","490776","12168","169","1"],["87178291200","1220496076800","3966612249600","5288816332800","3636061228800","1454424491520","363606122880","59364264960","6492966480","480960480","24048024","794976","16562","196","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","2","7","34","209","1546","13327","130922","1441729","17572114","234662231","3405357682","53334454417","896324308634","16083557845279"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-1","1"],["1","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":178714,"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":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-b41da8022f0b4d60"]},{"id":178715,"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-b41da8022f0b4d60"]},{"id":178716,"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-b41da8022f0b4d60"]}],"region_exact":{},"incoming_edges":[{"id":"e-b41da8022f0b4d60","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-0c345ddac20c06a3","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":"5501684eadce2c79ed9b65fc0cadbfcb1fc140ce29ae1f534ba177c8b50ac1f3","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-9e4edf560ed848c1","role":"sequence","name":"Triangle read by rows: the unsigned Lah numbers T(n, k) = binomial(n-1, k-1)*n!/k! if n > 0 and k > 0, T(n, 0) = 0^n and otherwise 0, for n >= 0 and 0 <= k <= n.","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]"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-37bf438676343978","kind":"fingerprint_related","node_a_id":"n-afa0b50cf12c7183","node_b_id":"n-0c345ddac20c06a3","fingerprint":"f6-f3c222c4860e5244","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f3c222c4860e5244"},"other_id":"n-afa0b50cf12c7183","other":{"id":"n-afa0b50cf12c7183","name":"derivative of Triangle read by rows: the unsigned Lah numbers T(n, k) = binomial(n-1, k-1)*n!/k! if n > 0 and k > 0, T(n, 0) = 0^n and otherwise 0, for n >= 0 and 0 <= k <= n.","first_index":1,"fingerprint":"f6-f3c222c4860e5244","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["1"],["2","2"]],"card_rows":[["1"],["2","2"],["6","12","3"],["24","72","36","4"]],"row_sums_preview":["0","1","4","21","136","1045","9276","93289"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-9e4edf560ed848c1","steps":["D"],"edges":["e-18cf672b6f5e39f5"],"text":"A271703 \u2192 D"},"headline":"A271703 \u2192 D","chain":null}},{"id":"l-44a030ef37a0ca55","kind":"fingerprint_related","node_a_id":"n-0c345ddac20c06a3","node_b_id":"n-44f7400f4634c690","fingerprint":"f6-f3c222c4860e5244","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f3c222c4860e5244"},"other_id":"n-44f7400f4634c690","other":{"id":"n-44f7400f4634c690","name":"derivative of Triangle read by rows: T(n, k) = 2^n * [x^k] hypergeom([-n, -n, 1], [1], x).","first_index":1,"fingerprint":"f6-f3c222c4860e5244","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["16","16"],["72","288","144"],["256","2304","4608","1536"]],"card_rows":[["16","16"],["72","288","144"],["256","2304","4608","1536"],["800","12800","57600","76800","19200"]],"row_sums_preview":["32","504","8704","167200","3561984","83586944","2145026048","59791385088"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-2ee8dc15b8ca5b2d","steps":["D"],"edges":["e-0c524bc9c76b4747"],"text":"A395972 \u2192 D"},"headline":"A395972 \u2192 D","chain":null}},{"id":"l-50f9acc7cd1db35a","kind":"fingerprint_related","node_a_id":"n-8ab17db79dad1880","node_b_id":"n-0c345ddac20c06a3","fingerprint":"f6-f3c222c4860e5244","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f3c222c4860e5244"},"other_id":"n-8ab17db79dad1880","other":{"id":"n-8ab17db79dad1880","name":"basis_map_A008277 of Triangle of unsigned Stirling numbers of the first kind (see A048994), read by rows, T(n,k) for 0 <= k <= n.","first_index":0,"fingerprint":"f6-f3c222c4860e5244","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","1"],["2","4","1"]],"card_rows":[["1"],["1","1"],["2","4","1"],["6","18","9","1"]],"row_sums_preview":["1","2","7","34","209","1546","13327","130922"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-b9fbfb57d9269e2f","steps":["basis A008277"],"edges":["e-07cfebc08bea7e3c"],"text":"A132393 \u2192 basis A008277"},"headline":"A132393 \u2192 basis A008277","chain":null}},{"id":"l-8cef8a2032300889","kind":"fingerprint_related","node_a_id":"n-2ee8dc15b8ca5b2d","node_b_id":"n-0c345ddac20c06a3","fingerprint":"f6-f3c222c4860e5244","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f3c222c4860e5244"},"other_id":"n-2ee8dc15b8ca5b2d","other":{"id":"n-2ee8dc15b8ca5b2d","name":"Triangle read by rows: T(n, k) = 2^n * [x^k] hypergeom([-n, -n, 1], [1], x).","first_index":1,"fingerprint":"f6-f3c222c4860e5244","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["4","16","8"],["8","72","144","48"],["16","256","1152","1536","384"]],"card_rows":[["4","16","8"],["8","72","144","48"],["16","256","1152","1536","384"],["32","800","6400","19200","19200","3840"]],"row_sums_preview":["28","272","3344","49472","852928","16758016","369082624","8996922368"],"regions":["(-inf, 0)","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-2ee8dc15b8ca5b2d","steps":[],"edges":[],"text":"A395972"},"headline":"Triangle read by rows: T(n, k) = 2^n * [x^k] hypergeom([-n, -n, 1], [1], x).","chain":null}},{"id":"l-ad50a41d597bac8a","kind":"fingerprint_related","node_a_id":"n-0c345ddac20c06a3","node_b_id":"n-e64be4d706cd3d64","fingerprint":"f6-f3c222c4860e5244","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f3c222c4860e5244"},"other_id":"n-e64be4d706cd3d64","other":{"id":"n-e64be4d706cd3d64","name":"theta_plus_one of Triangle read by rows: T(n,k) = binomial(n,k)*(n-1)!/(k-1)!.","first_index":1,"fingerprint":"f6-f3c222c4860e5244","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","2"],["6","12","3"]],"card_rows":[["1"],["2","2"],["6","12","3"],["24","72","36","4"]],"row_sums_preview":["1","4","21","136","1045","9276","93289","1047376"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-4a0fd05ab0621175","steps":["\u03b8+1"],"edges":["e-102bf30598aa8ffb"],"text":"A105278 \u2192 \u03b8+1"},"headline":"A105278 \u2192 \u03b8+1","chain":null}},{"id":"l-bad6ca4f09dad0f5","kind":"fingerprint_related","node_a_id":"n-0c345ddac20c06a3","node_b_id":"n-2aa2062a6ac07dd8","fingerprint":"f6-f3c222c4860e5244","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f3c222c4860e5244"},"other_id":"n-2aa2062a6ac07dd8","other":{"id":"n-2aa2062a6ac07dd8","name":"reversal of Triangle read by rows: T(n, k) = 2^n * [x^k] hypergeom([-n, -n, 1], [1], x).","first_index":1,"fingerprint":"f6-f3c222c4860e5244","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["8","16","4"],["48","144","72","8"],["384","1536","1152","256","16"]],"card_rows":[["8","16","4"],["48","144","72","8"],["384","1536","1152","256","16"],["3840","19200","19200","6400","800","32"]],"row_sums_preview":["28","272","3344","49472","852928","16758016","369082624","8996922368"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-2ee8dc15b8ca5b2d","steps":["reverse"],"edges":["e-72c2a78a0f17e32e"],"text":"A395972 \u2192 reverse"},"headline":"A395972 \u2192 reverse","chain":null}},{"id":"l-bdf187e3b21a379d","kind":"fingerprint_related","node_a_id":"n-0c345ddac20c06a3","node_b_id":"n-e5906904a5f133ef","fingerprint":"f6-f3c222c4860e5244","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f3c222c4860e5244"},"other_id":"n-e5906904a5f133ef","other":{"id":"n-e5906904a5f133ef","name":"derivative of T(n,k) is the number of partial bijections of height k (height(alpha) = |Im(alpha)|) of an n-element set.","first_index":0,"fingerprint":"f6-f3c222c4860e5244","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["1"],["4","4"]],"card_rows":[["1"],["4","4"],["9","36","18"],["16","144","288","96"]],"row_sums_preview":["0","1","8","63","544","5225","55656","653023"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-47fe395e9f7d5f86","steps":["D"],"edges":["e-4ba44c65cfbd3ffc"],"text":"A144084 \u2192 D"},"headline":"A144084 \u2192 D","chain":null}},{"id":"l-cd1c1cda6e09d312","kind":"fingerprint_related","node_a_id":"n-47fe395e9f7d5f86","node_b_id":"n-0c345ddac20c06a3","fingerprint":"f6-f3c222c4860e5244","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f3c222c4860e5244"},"other_id":"n-47fe395e9f7d5f86","other":{"id":"n-47fe395e9f7d5f86","name":"T(n,k) is the number of partial bijections of height k (height(alpha) = |Im(alpha)|) of an n-element set.","first_index":0,"fingerprint":"f6-f3c222c4860e5244","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","1"],["1","4","2"]],"card_rows":[["1"],["1","1"],["1","4","2"],["1","9","18","6"]],"row_sums_preview":["1","2","7","34","209","1546","13327","130922"],"regions":["(-inf, 0)","(-inf, 0]"],"region_exact":{},"oeis_ids":["A144084"],"derivation":{"root":"n-47fe395e9f7d5f86","steps":[],"edges":[],"text":"A144084"},"headline":"A144084: T(n,k) is the number of partial bijections of height k (height(alpha) = |Im(alpha)|) of an n-element set.","chain":null}},{"id":"l-fb96895fc5655385","kind":"fingerprint_related","node_a_id":"n-0c345ddac20c06a3","node_b_id":"n-4e758fcecfe3599a","fingerprint":"f6-f3c222c4860e5244","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f3c222c4860e5244"},"other_id":"n-4e758fcecfe3599a","other":{"id":"n-4e758fcecfe3599a","name":"theta_plus_one 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.","first_index":0,"fingerprint":"f6-f3c222c4860e5244","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","2"],["6","12","3"]],"card_rows":[["1"],["2","2"],["6","12","3"],["24","72","36","4"]],"row_sums_preview":["1","4","21","136","1045","9276","93289","1047376"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-efe23a76953cf69d","steps":["basis A008277","\u03b8+1"],"edges":["e-31e8b29857f291e5","e-8df45ffcd51dbfac"],"text":"A130534 \u2192 basis A008277 \u2192 \u03b8+1"},"headline":"A130534 \u2192 basis A008277 \u2192 \u03b8+1","chain":null}}],"oeis":[],"bfiles":[],"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:      1,       1;\n   3:      2,       4,       1;\n   4:      6,      18,       9,       1;\n   5:     24,      96,      72,      16,       1;\n   6:    120,     600,     600,     200,      25,      1;\n   7:    720,    4320,    5400,    2400,     450,     36,     1;\n   8:   5040,   35280,   52920,   29400,    7350,    882,    49,    1;\n   9:  40320,  322560,  564480,  376320,  117600,  18816,  1568,   64,  1;\n  10: 362880, 3265920, 6531840, 5080320, 1905120, 381024, 42336, 2592, 81, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (n-1+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[(n - 1 + 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] = (n - 1 + 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}}],"derivation":{"root":"n-9e4edf560ed848c1","steps":["D+1"],"edges":["e-b41da8022f0b4d60"],"text":"A271703 \u2192 D+1"},"headline":"A271703 \u2192 D+1","chain":null}
