{"id":"n-e41c30a965e7cb9c","revision":0,"first_index":0,"fingerprint":"f6-bd19e0a95c26bd62","name":"Exponential Riordan array [1, log((1-x)/(1-2x))].","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"],["0","1"],["0","3","1"],["0","14","9","1"],["0","90","83","18","1"],["0","744","870","275","30","1"],["0","7560","10474","4275","685","45","1"],["0","91440","143892","70924","14805","1435","63","1"],["0","1285200","2233356","1274196","324289","41160","2674","84","1"],["0","20603520","38769840","24870572","7398972","1151409","98280","4578","108","1"],["0","371226240","745785936","526791060","177945200","32559975","3419913","209790","7350","135","1"],["0","7428153600","15766233120","12072814776","4533332100","947490830","117466965","8889573","410850","11220","165","1"],["0","163459296000","363640940640","298275087528","122524830076","28637334990","4071519815","364669074","20830623","751410","16445","198","1"],["0","3923502105600","9092247615360","7914320990784","3512369295864","903331310596","144200769570","14852759207","1006107102","44919303","1299870","23309","234","1"],["0","102017281766400","245065185624960","224688938947200","106668976545768","29807441892228","5256847141546","609681627555","47591781809","2523519999","90483393","2147145","32123","273","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","1","4","24","192","1920","23040","322560","5160960","92897280","1857945600","40874803200","980995276800","25505877196800","714164561510400"],"nonn":true,"definitions":[{"position":0,"source":"oeis_catalog:A131222","source_kind":"oeis_catalog","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["0","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-3","1"],["3","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-4"],["6"],["-2"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":169403,"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-5195cbe6aa43ea8c"]},{"id":169404,"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-5195cbe6aa43ea8c"]},{"id":169405,"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-5195cbe6aa43ea8c"]},{"id":169406,"position":3,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"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 >= 1","region_exact":null,"edges":["e-5195cbe6aa43ea8c"]}],"region_exact":{},"incoming_edges":[{"id":"e-5195cbe6aa43ea8c","operation":{"operation":"theorem_check","theorem":"liu_wang_three_term"},"operation_detail":"liu_wang_three_term","conclusion_id":"n-e41c30a965e7cb9c","premise_count":0,"theorem":"L. L. Liu and Y. Wang, Adv. Appl. Math. 38 (2007) 542-560, Theorem 2.1 and Corollary 2.4 (A unified approach to polynomial sequences with only real zeros; the lag row as the interlacer, b_n = 0), weak interlacing of consecutive rows, with nonnegativity of the coefficients replaced, where a root region is certified, by an exact root-location induction","lean_route":"sign","lean_declarations":["RealRooted.threeTerm_interlaces_of_eval_nonpos","RealRooted.threeTerm_interlaces","RealRooted.eval_nonpos_of_natDegree_le_two"],"certificate_hash":"aae1723538f59ce804f01c43dfa3ee648e82ec5c0f53dcf6346f03cbb6f4f5e1","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":"previous_interlaces_next"}],"premises":[]}],"used_by":[{"id":"e-0f06a4dd5f61c0ed","operation":"basis_map","operation_detail":"A013611","conclusion_id":"n-cb6978de4b461a84","conclusion_name":"basis_map_A013611 of Exponential Riordan array [1, log((1-x)/(1-2x))]."},{"id":"e-603341b97b96be9a","operation":"basis_map","operation_detail":"A013610","conclusion_id":"n-e831bdf418cfc910","conclusion_name":"basis_map_A013610 of Exponential Riordan array [1, log((1-x)/(1-2x))]."},{"id":"e-a01a99b69c5df90f","operation":"basis_map","operation_detail":"A013609","conclusion_id":"n-3bc49e97c8fa21a8","conclusion_name":"basis_map_A013609 of Exponential Riordan array [1, log((1-x)/(1-2x))]."},{"id":"e-c5bcfc8ae4223484","operation":"basis_map","operation_detail":"A007318","conclusion_id":"n-b515d243eac1bbcc","conclusion_name":"shift_x_plus_one of Exponential Riordan array [1, log((1-x)/(1-2x))]."},{"id":"e-c92937f5d8a12889","operation":"basis_map","operation_detail":"A008277","conclusion_id":"n-3b33c24113e19b2d","conclusion_name":"basis_map_A008277 of Exponential Riordan array [1, log((1-x)/(1-2x))]."},{"id":"e-24705a6168ea5c39","operation":"preserver_image","operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-27f2c006ebd388c0","conclusion_name":"theta_plus_one of Exponential Riordan array [1, log((1-x)/(1-2x))]."},{"id":"e-7be6f1a3c1ffe0d4","operation":"preserver_image","operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-890b2f4d886ce308","conclusion_name":"derivative of Exponential Riordan array [1, log((1-x)/(1-2x))]."},{"id":"e-813b0037da3554e5","operation":"preserver_image","operation_detail":"reversal, shift 0","conclusion_id":"n-e395a1d44939f46e","conclusion_name":"reversal of Exponential Riordan array [1, log((1-x)/(1-2x))]."},{"id":"e-854bbc1bfe7bea67","operation":"preserver_image","operation_detail":"affine, shift 0","conclusion_id":"n-b515d243eac1bbcc","conclusion_name":"shift_x_plus_one of Exponential Riordan array [1, log((1-x)/(1-2x))]."},{"id":"e-b07c588e5f21b876","operation":"preserver_image","operation_detail":"mobius, shift 0","conclusion_id":"n-7a42f3124cafc007","conclusion_name":"mobius_x_over_x_plus_one of Exponential Riordan array [1, log((1-x)/(1-2x))]."},{"id":"e-ddbda95705f62f87","operation":"preserver_image","operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-3ca3dd82909bb758","conclusion_name":"theta_plus_one_squared of Exponential Riordan array [1, log((1-x)/(1-2x))]."},{"id":"e-f2fcdcfc29ef7dfb","operation":"preserver_image","operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-226363e48ade0e33","conclusion_name":"one_plus_derivative of Exponential Riordan array [1, log((1-x)/(1-2x))]."}],"used_by_total":12,"links":[{"id":"l-271c2b02d92353e7","kind":"fingerprint_related","node_a_id":"n-e41c30a965e7cb9c","node_b_id":"n-fd3aab150239f748","fingerprint":"f6-bd19e0a95c26bd62","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-bd19e0a95c26bd62"},"other_id":"n-fd3aab150239f748","other":{"id":"n-fd3aab150239f748","name":"reversal of Matrix product of unsigned Lah-triangle |A008297(n,k)| and unsigned Stirling1-triangle |A008275(n,k)|.","first_index":0,"fingerprint":"f6-bd19e0a95c26bd62","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","3"],["1","9","14"]],"card_rows":[["1"],["1","3"],["1","9","14"],["1","18","83","90"]],"row_sums_preview":["1","4","24","192","1920","23040","322560","5160960"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-6ace63f6e12c5046","steps":["reverse"],"edges":["e-bb6bec1a50d729ff"],"text":"A079638 \u2192 reverse"},"headline":"A079638 \u2192 reverse","chain":null}},{"id":"l-5326c20b295714a1","kind":"fingerprint_related","node_a_id":"n-6ace63f6e12c5046","node_b_id":"n-e41c30a965e7cb9c","fingerprint":"f6-bd19e0a95c26bd62","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-bd19e0a95c26bd62"},"other_id":"n-6ace63f6e12c5046","other":{"id":"n-6ace63f6e12c5046","name":"Matrix product of unsigned Lah-triangle |A008297(n,k)| and unsigned Stirling1-triangle |A008275(n,k)|.","first_index":0,"fingerprint":"f6-bd19e0a95c26bd62","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["3","1"],["14","9","1"]],"card_rows":[["1"],["3","1"],["14","9","1"],["90","83","18","1"]],"row_sums_preview":["1","4","24","192","1920","23040","322560","5160960"],"regions":["(-inf, +inf)","(-inf, 0)"],"region_exact":{},"oeis_ids":["A079638"],"derivation":{"root":"n-6ace63f6e12c5046","steps":[],"edges":[],"text":"A079638"},"headline":"A079638: Matrix product of unsigned Lah-triangle |A008297(n,k)| and unsigned Stirling1-triangle |A008275(n,k)|.","chain":null}},{"id":"l-8fe30d530eb437ad","kind":"fingerprint_related","node_a_id":"n-e41c30a965e7cb9c","node_b_id":"n-e395a1d44939f46e","fingerprint":"f6-bd19e0a95c26bd62","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-bd19e0a95c26bd62"},"other_id":"n-e395a1d44939f46e","other":{"id":"n-e395a1d44939f46e","name":"reversal of Exponential Riordan array [1, log((1-x)/(1-2x))].","first_index":0,"fingerprint":"f6-bd19e0a95c26bd62","polytool_family":null,"degree_pattern":"irregular","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1"],["1","3"]],"card_rows":[["1"],["1"],["1","3"],["1","9","14"]],"row_sums_preview":["1","1","4","24","192","1920","23040","322560"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-e41c30a965e7cb9c","steps":["reverse"],"edges":["e-813b0037da3554e5"],"text":"A131222 \u2192 reverse"},"headline":"A131222 \u2192 reverse","chain":null}},{"id":"l-38250b184e2cb0c9","kind":"oeis_match","node_a_id":"n-e41c30a965e7cb9c","node_b_id":null,"fingerprint":null,"oeis_id":"A002866","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A002866","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-5395d44008d2e710","kind":"oeis_match","node_a_id":"n-e41c30a965e7cb9c","node_b_id":null,"fingerprint":null,"oeis_id":"A052656","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":3,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":3,"target":"row_sums","terms_compared":12},"in_catalog":false,"oeis_id":"A052656","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-65d4c6fbfae70bfc","kind":"oeis_match","node_a_id":"n-e41c30a965e7cb9c","node_b_id":null,"fingerprint":null,"oeis_id":"A121892","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":2,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":2,"target":"row_sums","terms_compared":11},"in_catalog":false,"oeis_id":"A121892","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-68bd7d6d39a7283d","kind":"oeis_match","node_a_id":"n-e41c30a965e7cb9c","node_b_id":null,"fingerprint":null,"oeis_id":"A131222","in_catalog":1,"target":"triangle","row_alignment":"tabl","rows_matched":9,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":9,"target":"triangle","terms_compared":45},"in_catalog":true,"oeis_id":"A131222","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-7189410198b3e4c9","kind":"oeis_match","node_a_id":"n-e41c30a965e7cb9c","node_b_id":null,"fingerprint":null,"oeis_id":"A052594","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":3,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":3,"target":"row_sums","terms_compared":12},"in_catalog":false,"oeis_id":"A052594","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-822ed45ac29d8c38","kind":"oeis_match","node_a_id":"n-e41c30a965e7cb9c","node_b_id":null,"fingerprint":null,"oeis_id":"A032107","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":2,"low_zeros_dropped":false,"oeis_offset":1,"oeis_term_position":1,"target":"row_sums","terms_compared":13},"in_catalog":false,"oeis_id":"A032107","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-b0bed81db853c9dd","kind":"oeis_match","node_a_id":"n-e41c30a965e7cb9c","node_b_id":null,"fingerprint":null,"oeis_id":"A079638","in_catalog":1,"target":"triangle","row_alignment":"tabl","rows_matched":8,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":true,"oeis_offset":1,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":8,"target":"triangle","terms_compared":43},"in_catalog":true,"oeis_id":"A079638","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A131222","relation":"identification"},{"oeis_id":"A002866","relation":"match_row_sums"},{"oeis_id":"A032107","relation":"match_row_sums"},{"oeis_id":"A052594","relation":"match_row_sums"},{"oeis_id":"A052656","relation":"match_row_sums"},{"oeis_id":"A121892","relation":"match_row_sums"},{"oeis_id":"A079638","relation":"match_triangle"},{"oeis_id":"A131222","relation":"match_triangle"}],"bfiles":[{"node_id":"n-e41c30a965e7cb9c","oeis_id":"A002866","target":"row_sums","oeis_offset":0,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":21,"oeis_terms_available":21,"term_count":200,"raw_size":40000},{"node_id":"n-e41c30a965e7cb9c","oeis_id":"A079638","target":"triangle","oeis_offset":1,"first_row":1,"rows_used":44,"terms_matched":43,"oeis_terms_agreeing":43,"oeis_terms_available":43,"term_count":990,"raw_size":32203},{"node_id":"n-e41c30a965e7cb9c","oeis_id":"A131222","target":"triangle","oeis_offset":0,"first_row":0,"rows_used":44,"terms_matched":45,"oeis_terms_agreeing":45,"oeis_terms_available":45,"term_count":990,"raw_size":30292}],"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: 0,        1;\n  2: 0,        3,        1;\n  3: 0,       14,        9,        1;\n  4: 0,       90,       83,       18,       1;\n  5: 0,      744,      870,      275,      30,       1;\n  6: 0,     7560,    10474,     4275,     685,      45,     1;\n  7: 0,    91440,   143892,    70924,   14805,    1435,    63,    1;\n  8: 0,  1285200,  2233356,  1274196,  324289,   41160,  2674,   84,   1;\n  9: 0, 20603520, 38769840, 24870572, 7398972, 1151409, 98280, 4578, 108, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (3*n-3+x)*P(n-1,x) - (2*n^2-6*n+4)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := x;\nP[n_, x_] := P[n, x] = Expand[(3*n - 3 + x)*P[n - 1, x] - (2*n^2 - 6*n + 4)*P[n - 2, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = x; for(n = 2, N, my(i = n + 1); v[i] = (3*n - 3 + x)*v[i - 1] - (2*n^2 - 6*n + 4)*v[i - 2]); 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}},{"oeis_id":"A079638","target":"triangle","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"low","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:         1;\n   2:         3,         1;\n   3:        14,         9,         1;\n   4:        90,        83,        18,         1;\n   5:       744,       870,       275,        30,        1;\n   6:      7560,     10474,      4275,       685,       45,       1;\n   7:     91440,    143892,     70924,     14805,     1435,      63,      1;\n   8:   1285200,   2233356,   1274196,    324289,    41160,    2674,     84,    1;\n   9:  20603520,  38769840,  24870572,   7398972,  1151409,   98280,   4578,  108,   1;\n  10: 371226240, 745785936, 526791060, 177945200, 32559975, 3419913, 209790, 7350, 135, 1;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in ascending powers of x, starting at the lowest nonzero coefficient, where P(n,x) = (3*n-3+x)*P(n-1,x) - (2*n^2-6*n+4)*P(n-2,x) for n >= 3, with P(1,x) = x, P(2,x) = 3*x + x^2.","mathematica":"P[1, x_] := x;\nP[2, x_] := 3*x + x^2;\nP[n_, x_] := P[n, x] = Expand[(3*n - 3 + x)*P[n - 1, x] - (2*n^2 - 6*n + 4)*P[n - 2, x]];\nTable[PadRight[CoefficientList[Expand[P[n, x]/x^Exponent[P[n, x], x, Min]], x], n], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = x; v[2] = 3*x + x^2; for(n = 3, N, my(i = n); v[i] = (3*n - 3 + x)*v[i - 1] - (2*n^2 - 6*n + 4)*v[i - 2]); v};\nrow(n, p) = {my(c = Vecrev(p/x^valuation(p, x))); c = concat(c, vector(n - #c)); 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":"low","regular_triangle":true}},{"oeis_id":"A131222","target":"triangle","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0: 1;\n  1: 0,        1;\n  2: 0,        3,        1;\n  3: 0,       14,        9,        1;\n  4: 0,       90,       83,       18,       1;\n  5: 0,      744,      870,      275,      30,       1;\n  6: 0,     7560,    10474,     4275,     685,      45,     1;\n  7: 0,    91440,   143892,    70924,   14805,    1435,    63,    1;\n  8: 0,  1285200,  2233356,  1274196,  324289,   41160,  2674,   84,   1;\n  9: 0, 20603520, 38769840, 24870572, 7398972, 1151409, 98280, 4578, 108, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (3*n-3+x)*P(n-1,x) - (2*n^2-6*n+4)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := x;\nP[n_, x_] := P[n, x] = Expand[(3*n - 3 + x)*P[n - 1, x] - (2*n^2 - 6*n + 4)*P[n - 2, x]];\nTable[PadRight[CoefficientList[P[n, x], x], n + 1], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = x; for(n = 2, N, my(i = n + 1); v[i] = (3*n - 3 + x)*v[i - 1] - (2*n^2 - 6*n + 4)*v[i - 2]); v};\nrow(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); 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":true}}],"derivation":{"root":"n-e41c30a965e7cb9c","steps":[],"edges":[],"text":"A131222"},"headline":"A131222: Exponential Riordan array [1, log((1-x)/(1-2x))].","chain":null}