{"id":"n-3dbfcbca0c0c8d13","revision":1,"first_index":0,"fingerprint":"f6-b9857b4c4b3ea9b8","name":"Triangle read by rows: T(n,k) (n >= 1, 0 <= k <= ceiling(n/2)-1) = number of permutations of [n] with k peaks.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"half_affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["2"],["4","2"],["8","16"],["16","88","16"],["32","416","272"],["64","1824","2880","272"],["128","7680","24576","7936"],["256","31616","185856","137216","7936"],["512","128512","1304832","1841152","353792"],["1024","518656","8728576","21253376","9061376","353792"],["2048","2084864","56520704","222398464","175627264","22368256"],["4096","8361984","357888000","2174832640","2868264960","795300864","22368256"],["8192","33497088","2230947840","20261765120","41731645440","21016670208","1903757312"],["16384","134094848","13754155008","182172651520","559148810240","460858269696","89702612992","1903757312"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","2","6","24","120","720","5040","40320","362880","3628800","39916800","479001600","6227020800","87178291200","1307674368000"],"nonn":true,"definitions":[{"position":0,"source":"oeis_catalog:A008303","source_kind":"oeis_catalog","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["2","-1"],["0","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","2","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":159969,"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-00ceaf3777abe095"]},{"id":159970,"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-00ceaf3777abe095"]},{"id":159971,"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-00ceaf3777abe095"]}],"region_exact":{},"incoming_edges":[{"id":"e-00ceaf3777abe095","operation":{"index_shift":1,"operation":"gamma_polynomial","scale":"1/4"},"operation_detail":"scale 1/4, shift 1","conclusion_id":"n-3dbfcbca0c0c8d13","premise_count":1,"theorem":"gamma criterion (RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff): if P = sum_j gamma_j x^j (1 + x)^(d - 2j) with deg Gamma <= d/2, then P is nonzero and real-rooted with roots in (-inf, 0] if and only if Gamma(z) = sum_j gamma_j z^j is; Gamma(c z) with c != 0 has the roots of Gamma divided by c","lean_route":null,"lean_declarations":[],"certificate_hash":"2b96c6aae415fe3c6e041772f7eb71fe49be1b6ebbd52b2c4e92dfa0ec98c02b","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-ff270bcb802a85af","role":"sequence","name":"eulerian","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":[{"id":"e-4f4a07a2cead1a74","operation":"preserver_image","operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-db03f4043335d472","conclusion_name":"derivative of Triangle read by rows: T(n,k) (n >= 1, 0 <= k <= ceiling(n/2)-1) = number of permutations of [n] with k peaks."},{"id":"e-71d052822858573a","operation":"preserver_image","operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-37c02f0876082565","conclusion_name":"theta_plus_one of Triangle read by rows: T(n,k) (n >= 1, 0 <= k <= ceiling(n/2)-1) = number of permutations of [n] with k peaks."},{"id":"e-8621fad14db1f00c","operation":"preserver_image","operation_detail":"affine, shift 0","conclusion_id":"n-3058adfe9873c66a","conclusion_name":"shift_x_plus_one of Triangle read by rows: T(n,k) (n >= 1, 0 <= k <= ceiling(n/2)-1) = number of permutations of [n] with k peaks."},{"id":"e-e126124009b7b91d","operation":"preserver_image","operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-74461042f186bb00","conclusion_name":"theta_plus_one_squared of Triangle read by rows: T(n,k) (n >= 1, 0 <= k <= ceiling(n/2)-1) = number of permutations of [n] with k peaks."},{"id":"e-ea99ed3c2ecebb89","operation":"preserver_image","operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-78f78f6ebc7032b5","conclusion_name":"one_plus_derivative of Triangle read by rows: T(n,k) (n >= 1, 0 <= k <= ceiling(n/2)-1) = number of permutations of [n] with k peaks."}],"used_by_total":5,"links":[{"id":"l-7f7e10fe59e98699","kind":"fingerprint_related","node_a_id":"n-92480a234eee36bb","node_b_id":"n-3dbfcbca0c0c8d13","fingerprint":"f6-b9857b4c4b3ea9b8","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b9857b4c4b3ea9b8"},"other_id":"n-92480a234eee36bb","other":{"id":"n-92480a234eee36bb","name":"peak","first_index":1,"fingerprint":"f6-b9857b4c4b3ea9b8","polytool_family":"peak","degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2"],["4","2"]],"card_rows":[["1"],["2"],["4","2"],["8","16"]],"row_sums_preview":["1","2","6","24","120","720","5040","40320"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ff270bcb802a85af","steps":["\u03b3 (scale 1/4)"],"edges":["e-8b7786e9ad6af8fd"],"text":"A008292 \u2192 \u03b3 (scale 1/4)"},"headline":"A008292 \u2192 \u03b3 (scale 1/4)","chain":null}},{"id":"l-1e467f47aada520e","kind":"oeis_match","node_a_id":"n-3dbfcbca0c0c8d13","node_b_id":null,"fingerprint":null,"oeis_id":"A104150","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":2,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A104150","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-3b5f739e840b817c","kind":"oeis_match","node_a_id":"n-3dbfcbca0c0c8d13","node_b_id":null,"fingerprint":null,"oeis_id":"A159333","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":0,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":-2,"oeis_term_position":3,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A159333","oeis_nonn":false},"other_id":null,"other":null},{"id":"l-6be7a69806c1e1f2","kind":"oeis_match","node_a_id":"n-3dbfcbca0c0c8d13","node_b_id":null,"fingerprint":null,"oeis_id":"A338112","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":2,"target":"row_sums","terms_compared":13},"in_catalog":false,"oeis_id":"A338112","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-8f01afee88e8dc7b","kind":"oeis_match","node_a_id":"n-3dbfcbca0c0c8d13","node_b_id":null,"fingerprint":null,"oeis_id":"A358185","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":0,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":3,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A358185","oeis_nonn":false},"other_id":null,"other":null},{"id":"l-b19b1acf6e70ab20","kind":"oeis_match","node_a_id":"n-3dbfcbca0c0c8d13","node_b_id":null,"fingerprint":null,"oeis_id":"A354074","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":"A354074","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-ba14e3da01bb13d9","kind":"oeis_match","node_a_id":"n-3dbfcbca0c0c8d13","node_b_id":null,"fingerprint":null,"oeis_id":"A000142","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":1,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A000142","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-c9a2999d1de73d5e","kind":"oeis_match","node_a_id":"n-3dbfcbca0c0c8d13","node_b_id":null,"fingerprint":null,"oeis_id":"A124355","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":1,"oeis_term_position":2,"target":"row_sums","terms_compared":14},"in_catalog":false,"oeis_id":"A124355","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-df80a86d6a97d8f5","kind":"oeis_match","node_a_id":"n-3dbfcbca0c0c8d13","node_b_id":null,"fingerprint":null,"oeis_id":"A008303","in_catalog":1,"target":"triangle","row_alignment":"node_lengths","rows_matched":12,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":1,"oeis_term_position":0,"row_alignment":"node_lengths","rows_matched":12,"target":"triangle","terms_compared":42},"in_catalog":true,"oeis_id":"A008303","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-f5a609550f6a5541","kind":"oeis_match","node_a_id":"n-3dbfcbca0c0c8d13","node_b_id":null,"fingerprint":null,"oeis_id":"A370360","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":1,"oeis_term_position":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A370360","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A008303","relation":"identification"},{"oeis_id":"A000142","relation":"match_row_sums"},{"oeis_id":"A104150","relation":"match_row_sums"},{"oeis_id":"A124355","relation":"match_row_sums"},{"oeis_id":"A159333","relation":"match_row_sums"},{"oeis_id":"A338112","relation":"match_row_sums"},{"oeis_id":"A354074","relation":"match_row_sums"},{"oeis_id":"A358185","relation":"match_row_sums"},{"oeis_id":"A370360","relation":"match_row_sums"},{"oeis_id":"A008303","relation":"match_triangle"}],"bfiles":[{"node_id":"n-3dbfcbca0c0c8d13","oeis_id":"A008303","target":"triangle","oeis_offset":1,"first_row":0,"rows_used":62,"terms_matched":42,"oeis_terms_agreeing":42,"oeis_terms_available":42,"term_count":992,"raw_size":48905}],"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:   2;\n  2:   4,      2;\n  3:   8,     16;\n  4:  16,     88,      16;\n  5:  32,    416,     272;\n  6:  64,   1824,    2880,     272;\n  7: 128,   7680,   24576,    7936;\n  8: 256,  31616,  185856,  137216,   7936;\n  9: 512, 128512, 1304832, 1841152, 353792;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2+(n-1)*x)*P(n-1,x) + (2*x-2*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(2 + (n - 1)*x)*P[n - 1, x] + (2*x - 2*x^2)*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (2 + (n - 1)*x)*v[i - 1] + (2*x - 2*x^2)*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 - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}},{"oeis_id":"A008303","target":"triangle","first_row":0,"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:   2;\n   3:   4,      2;\n   4:   8,     16;\n   5:  16,     88,      16;\n   6:  32,    416,     272;\n   7:  64,   1824,    2880,     272;\n   8: 128,   7680,   24576,    7936;\n   9: 256,  31616,  185856,  137216,   7936;\n  10: 512, 128512, 1304832, 1841152, 353792;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2+(n-2)*x)*P(n-1,x) + (2*x-2*x^2)*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[(2 + (n - 2)*x)*P[n - 1, x] + (2*x - 2*x^2)*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] = (2 + (n - 2)*x)*v[i - 1] + (2*x - 2*x^2)*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":0,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-ff270bcb802a85af","steps":["\u03b3 (scale 1/4) (shift +1)"],"edges":["e-00ceaf3777abe095"],"text":"A008292 \u2192 \u03b3 (scale 1/4) (shift +1)"},"headline":"A008303: Triangle read by rows: T(n,k) (n >= 1, 0 <= k <= ceiling(n/2)-1) = number of permutations of [n] with k peaks.","chain":"A008292 \u2192 \u03b3 (scale 1/4) (shift +1)"}
