{"id":"n-82c964c6f2d99674","revision":0,"first_index":0,"fingerprint":"f6-72ae8e6419635e63","name":"derivative of theta_plus_one of Exponential Riordan array [x*exp(x),x].","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-2,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[[],[],["4"],["12","18"],["24","72","48"],["40","180","240","100"],["60","360","720","600","180"],["84","630","1680","2100","1260","294"],["112","1008","3360","5600","5040","2352","448"],["144","1512","6048","12600","15120","10584","4032","648"],["180","2160","10080","25200","37800","35280","20160","6480","900"],["220","2970","15840","46200","83160","97020","73920","35640","9900","1210"],["264","3960","23760","79200","166320","232848","221760","142560","59400","14520","1584"],["312","5148","34320","128700","308880","504504","576576","463320","257400","94380","20592","2028"],["364","6552","48048","200200","540540","1009008","1345344","1297296","900900","440440","144144","28392","2548"]],"degrees":[-1,-1,0,1,2,3,4,5,6,7,8,9,10,11,12],"row_sums":["0","0","4","30","144","560","1920","6048","17920","50688","138240","366080","946176","2396160","5963776"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["0"],["4"]],"recurrence":{"denominator":{"coeffs":[["-4","2"],["2","-3"],["0","1"]]},"terms":[{"coeff":{"coeffs":[["0","0","0"],["2","2","0"],["0","1","1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":173880,"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-c5c74718b95fb94d"]},{"id":173881,"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-c5c74718b95fb94d"]},{"id":173882,"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-c5c74718b95fb94d"]}],"region_exact":{},"incoming_edges":[{"id":"e-c5c74718b95fb94d","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-82c964c6f2d99674","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":"686dc5df3778077842b7e938aedbc042d9caaeb1d0e5113b8fa8539336883d75","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-d86f0fa1a60cdcc7","role":"sequence","name":"theta_plus_one of Exponential Riordan array [x*exp(x),x].","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-1b084a27f6281ee7","kind":"fingerprint_related","node_a_id":"n-82c964c6f2d99674","node_b_id":"n-04aed1ba138cb459","fingerprint":"f6-72ae8e6419635e63","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-72ae8e6419635e63"},"other_id":"n-04aed1ba138cb459","other":{"id":"n-04aed1ba138cb459","name":"derivative of reversal of Triangle read by rows: T(n,k) = k*(k+1)*binomial(n,k)/2 (1 <= k <= n).","first_index":0,"fingerprint":"f6-72ae8e6419635e63","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["2"],["9","6"]],"card_rows":[["2"],["9","6"],["24","36","12"],["50","120","90","20"]],"row_sums_preview":["0","2","15","72","280","960","3024","8960"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-787407680a56b31c","steps":["reverse","D"],"edges":["e-03e6ee68811b0a8b","e-f11652baa9c3035f"],"text":"A124932 \u2192 reverse \u2192 D"},"headline":"A124932 \u2192 reverse \u2192 D","chain":null}},{"id":"l-1bcd30d92fad6428","kind":"fingerprint_related","node_a_id":"n-2c11b65ec5b65118","node_b_id":"n-82c964c6f2d99674","fingerprint":"f6-72ae8e6419635e63","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-72ae8e6419635e63"},"other_id":"n-2c11b65ec5b65118","other":{"id":"n-2c11b65ec5b65118","name":"reversal of Triangle read by rows: T(n,k) = k*(k+1)*binomial(n,k)/2 (1 <= k <= n).","first_index":0,"fingerprint":"f6-72ae8e6419635e63","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["3","2"],["6","9","3"]],"card_rows":[["1"],["3","2"],["6","9","3"],["10","24","18","4"]],"row_sums_preview":["1","5","18","56","160","432","1120","2816"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-787407680a56b31c","steps":["reverse"],"edges":["e-03e6ee68811b0a8b"],"text":"A124932 \u2192 reverse"},"headline":"A124932 \u2192 reverse","chain":null}},{"id":"l-2bdb0ddd1e457df8","kind":"fingerprint_related","node_a_id":"n-f889074884265bc4","node_b_id":"n-82c964c6f2d99674","fingerprint":"f6-72ae8e6419635e63","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-72ae8e6419635e63"},"other_id":"n-f889074884265bc4","other":{"id":"n-f889074884265bc4","name":"derivative of Riordan array (1-x,x/(1-x)).","first_index":0,"fingerprint":"f6-72ae8e6419635e63","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["0","2"],["0","2","3"],["0","2","6","4"]],"card_rows":[["0","2"],["0","2","3"],["0","2","6","4"],["0","2","9","12","5"]],"row_sums_preview":["2","5","12","28","64","144","320","704"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7717563017cb9e5f","steps":["D"],"edges":["e-23d1d4324cad1a70"],"text":"A159854 \u2192 D"},"headline":"A159854 \u2192 D","chain":null}},{"id":"l-73ab2e5243f81d4e","kind":"fingerprint_related","node_a_id":"n-6474b315bc694992","node_b_id":"n-82c964c6f2d99674","fingerprint":"f6-72ae8e6419635e63","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-72ae8e6419635e63"},"other_id":"n-6474b315bc694992","other":{"id":"n-6474b315bc694992","name":"derivative of Triangle read by rows: n-th row = unsigned coefficients of the characteristic polynomials of an n X n matrix with 2's on the diagonal and 1's elsewhere.","first_index":0,"fingerprint":"f6-72ae8e6419635e63","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["2"],["4","6"]],"card_rows":[["2"],["4","6"],["6","18","12"],["8","36","48","20"]],"row_sums_preview":["0","2","10","36","112","320","864","2240"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-801895951aa20f29","steps":["D"],"edges":["e-a1328219e60a28e7"],"text":"A103406 \u2192 D"},"headline":"A103406 \u2192 D","chain":null}},{"id":"l-8f4ea03448167c4d","kind":"fingerprint_related","node_a_id":"n-82c964c6f2d99674","node_b_id":"n-6a4c6a9fa7126307","fingerprint":"f6-72ae8e6419635e63","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-72ae8e6419635e63"},"other_id":"n-6a4c6a9fa7126307","other":{"id":"n-6a4c6a9fa7126307","name":"basis_map_A013609 of Array T(m,n) read by ascending antidiagonals: T(m,n) = m*binomial(n+m-2, n-1) for m, n >= 1.","first_index":0,"fingerprint":"f6-72ae8e6419635e63","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["3","2"],["8","12","4"]],"card_rows":[["1"],["3","2"],["8","12","4"],["20","48","36","8"]],"row_sums_preview":["1","5","24","112","512","2304","10240","45056"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-f24dc5f77f658331","steps":["basis A013609"],"edges":["e-37f97d572063199e"],"text":"A093375 \u2192 basis A013609"},"headline":"A093375 \u2192 basis A013609","chain":null}},{"id":"l-f13c59149a8afff0","kind":"fingerprint_related","node_a_id":"n-787407680a56b31c","node_b_id":"n-82c964c6f2d99674","fingerprint":"f6-72ae8e6419635e63","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-72ae8e6419635e63"},"other_id":"n-787407680a56b31c","other":{"id":"n-787407680a56b31c","name":"Triangle read by rows: T(n,k) = k*(k+1)*binomial(n,k)/2 (1 <= k <= n).","first_index":0,"fingerprint":"f6-72ae8e6419635e63","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","3"],["3","9","6"]],"card_rows":[["1"],["2","3"],["3","9","6"],["4","18","24","10"]],"row_sums_preview":["1","5","18","56","160","432","1120","2816"],"regions":["(-inf, 0]","(-inf, 0)"],"region_exact":{},"oeis_ids":["A124932"],"derivation":{"root":"n-787407680a56b31c","steps":[],"edges":[],"text":"A124932"},"headline":"A124932: Triangle read by rows: T(n,k) = k*(k+1)*binomial(n,k)/2 (1 <= k <= n).","chain":null}},{"id":"l-fb8133859e6c1626","kind":"fingerprint_related","node_a_id":"n-82c964c6f2d99674","node_b_id":"n-727a5b76084f597f","fingerprint":"f6-72ae8e6419635e63","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-72ae8e6419635e63"},"other_id":"n-727a5b76084f597f","other":{"id":"n-727a5b76084f597f","name":"basis_map_A013609 of one_plus_derivative of Exponential Riordan array [x*exp(x),x].","first_index":0,"fingerprint":"f6-72ae8e6419635e63","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["1"],["6","4"]],"card_rows":[["1"],["6","4"],["24","36","12"],["80","192","144","32"]],"row_sums_preview":["0","1","10","72","448","2560","13824","71680"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-f24dc5f77f658331","steps":["D","basis A013609"],"edges":["e-2695bc03a925fb68","e-6f5ba453aaec2135"],"text":"A093375 \u2192 D \u2192 basis A013609"},"headline":"A093375 \u2192 D \u2192 basis A013609","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:   0;\n  2:   4;\n  3:  12,   18;\n  4:  24,   72,   48;\n  5:  40,  180,  240,   100;\n  6:  60,  360,  720,   600,   180;\n  7:  84,  630, 1680,  2100,  1260,   294;\n  8: 112, 1008, 3360,  5600,  5040,  2352,  448;\n  9: 144, 1512, 6048, 12600, 15120, 10584, 4032, 648;\n","recurrence_sentence":"The row polynomials satisfy (2*n-4+(n^2-3*n+2)*x)*P(n,x) = (2*n+(n^2+2*n)*x+n^2*x^2)*P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 4.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 0;\nP[2, x_] := 4;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n + (n^2 + 2*n)*x + n^2*x^2)*P[n - 1, x])/(2*n - 4 + (n^2 - 3*n + 2)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 0; v[3] = 4; for(n = 3, N, my(i = n + 1); v[i] = ((2*n + (n^2 + 2*n)*x + n^2*x^2)*v[i - 1])/(2*n - 4 + (n^2 - 3*n + 2)*x)); 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-308cf60736924b4c","steps":["\u03b8+1","D"],"edges":["e-0b4ed87df247fda0","e-c5c74718b95fb94d"],"text":"A216973 \u2192 \u03b8+1 \u2192 D"},"headline":"A216973 \u2192 \u03b8+1 \u2192 D","chain":null}