{"id":"n-6e6daac86c9463fd","revision":0,"first_index":1,"fingerprint":"f6-b00b790aa67df404","name":"reversal of Coefficients of the derivatives of the Eulerian polynomials (with indexing as in A173018).","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":[["3","22","11"],["4","78","132","26"],["5","228","906","604","57"],["6","600","4764","7248","2382","120"],["7","1482","21465","62476","46857","8586","247"],["8","3514","87648","441170","624760","264702","29216","502"],["9","8104","334880","2731152","6551770","5241416","1365576","95680","1013"],["10","18324","1221096","15424416","58428684","78621240","38952456","6610464","305274","2036"],["11","40830","4304439","81501480","464229318","975073716","812561430","265273896","30563055","956542","4083"],["12","89958","14797260","409801050","3386252280","10539350556","13651032024","7528107540","1693126140","136600350","2959452","8178"],["13","196428","49910454","1984107860","23145679575","102746102904","195798232644","167827056552","64216314315","10286968700","595232358","9074628","16369"],["14","425776","165896868","9329000032","150412295210","924992990352","2491100787288","3132771722304","1868325590466","513884994640","60164918084","2544272736","27649478","32752"],["15","917266","545125685","42865023060","939215624039","7821155182990","28867348603413","51062391403800","44679592478325","19244899068942","3910577591495","341532954196","10716255765","83865490","65519"],["16","1965810","1774934280","193380192850","5680239612720","62890211696202","310556529483880","748235008668330","919123045268400","581960562297590","186333917690328","28586459861910","1893413204240","44626198350","253562040","131054"],["17","4194000","5736598860","859646383400","33475287268800","485488136976864","3145967817380300","10067099679152280","16788470920025850","14923085262245200","7046969775406596","1715982445843800","202286723740360","10300088390400","184209939300","764879848","262125"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["36","240","1800","15120","141120","1451520","16329600","199584000","2634508800","37362124800","566658892800","9153720576000","156920924160000","2845499424768000","54420176498688000"],"nonn":true,"definitions":[{"position":0,"source":"derived:reversal","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["3","22","11"],["4","78","132","26"]],"recurrence":{"denominator":{"coeffs":[["1"],["1"]]},"terms":[{"coeff":{"coeffs":[["-2","0","-2"],["-3","-2","-3"],["-1","-3","-1"],["0","-1","0"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","-2","0","2"],["0","-3","2","1"],["0","-1","1","0"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["2","4"],["3","4"],["1","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1","-1"],["0","1","-1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":188898,"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-0d05227aa568e6cc"]},{"id":188899,"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-0d05227aa568e6cc"]},{"id":188900,"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-0d05227aa568e6cc"]}],"region_exact":{},"incoming_edges":[{"id":"e-0d05227aa568e6cc","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"reversal\",\"schedule\":{\"first_index\":1,\"first_degree\":2,\"step\":1}}"},"operation_detail":"reversal, shift 0","conclusion_id":"n-6e6daac86c9463fd","premise_count":1,"theorem":"a real affine substitution x -> a x + b (a != 0), the reversal x^d p(1/x) (d >= deg p) and a real Mobius map (c x + e)^d p((a x + b)/(c x + e)) (a e - b c != 0, d >= deg p) map real roots to real roots: a root r of p gives the root (r - b)/a, 1/r (and 0) or (e r - b)/(a - c r) (and -e/c); for an input row that is a polynomial the degree condition holds automatically","lean_route":"reversed","lean_declarations":["RealRooted.X_pow_mul_reverse_eq_zero_or_splits","RealRooted.reflect_eq_zero_or_splits"],"certificate_hash":"b263c0f33ce3f3bd233b16dfeb376b843163346b14816ef6055ac749bed2568a","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-1e9cff7ed448d4e0","role":"sequence","name":"Coefficients of the derivatives of the Eulerian polynomials (with indexing as in A173018).","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-68a5ab7039a65575","kind":"fingerprint_related","node_a_id":"n-c186a5d9b6da5907","node_b_id":"n-6e6daac86c9463fd","fingerprint":"f6-b00b790aa67df404","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b00b790aa67df404"},"other_id":"n-c186a5d9b6da5907","other":{"id":"n-c186a5d9b6da5907","name":"derivative of row_factor parent of Triangle read by rows: T(n,k), for k=-n..n-1, is the scaled (by 2^n n!) probability that the sum of n uniform [-1, 1] variables is between k and k+1.","first_index":0,"fingerprint":"f6-b00b790aa67df404","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["1"],["4","2"]],"card_rows":[["1"],["4","2"],["11","22","3"],["26","132","78","4"]],"row_sums_preview":["0","1","6","36","240","1800","15120","141120"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-492f384fe03c9268","steps":["D"],"edges":["e-17fac32bb082f5ca"],"text":"row-factor cofactor of A101842 \u2192 D"},"headline":"row-factor cofactor of A101842 \u2192 D","chain":null}},{"id":"l-6b53fdc81a75b512","kind":"fingerprint_related","node_a_id":"n-84d0c5dbe02497ed","node_b_id":"n-6e6daac86c9463fd","fingerprint":"f6-b00b790aa67df404","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b00b790aa67df404"},"other_id":"n-84d0c5dbe02497ed","other":{"id":"n-84d0c5dbe02497ed","name":"derivative of eulerian","first_index":1,"fingerprint":"f6-b00b790aa67df404","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["1"],["4","2"]],"card_rows":[["1"],["4","2"],["11","22","3"],["26","132","78","4"]],"row_sums_preview":["0","1","6","36","240","1800","15120","141120"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ff270bcb802a85af","steps":["D"],"edges":["e-0215d14e0b41ed17"],"text":"A008292 \u2192 D"},"headline":"A008292 \u2192 D","chain":null}},{"id":"l-79966ed72e454dd3","kind":"fingerprint_related","node_a_id":"n-627ac699506485c3","node_b_id":"n-6e6daac86c9463fd","fingerprint":"f6-b00b790aa67df404","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b00b790aa67df404"},"other_id":"n-627ac699506485c3","other":{"id":"n-627ac699506485c3","name":"derivative of Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 2*x + 2.","first_index":0,"fingerprint":"f6-b00b790aa67df404","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["2"],["16","8"]],"card_rows":[["2"],["16","8"],["88","176","24"],["416","2112","1248","64"]],"row_sums_preview":["0","2","24","288","3840","57600","967680","18063360"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-41809911e05bb4ba","steps":["D"],"edges":["e-f6269d66468128f6"],"text":"A257609 \u2192 D"},"headline":"A257609 \u2192 D","chain":null}},{"id":"l-830a8449eea8e68b","kind":"fingerprint_related","node_a_id":"n-08acbfdfd91c4faa","node_b_id":"n-6e6daac86c9463fd","fingerprint":"f6-b00b790aa67df404","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b00b790aa67df404"},"other_id":"n-08acbfdfd91c4faa","other":{"id":"n-08acbfdfd91c4faa","name":"derivative of Euler's triangle: triangle of Eulerian numbers T(n,k) (n>=0, 0 <= k <= n) read by rows.","first_index":1,"fingerprint":"f6-b00b790aa67df404","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],[],["1"]],"card_rows":[["1"],["4","2"],["11","22","3"],["26","132","78","4"]],"row_sums_preview":["0","0","1","6","36","240","1800","15120"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-efd1fdf07e5468d8","steps":["D"],"edges":["e-8d8e49a698c1586a"],"text":"A173018 \u2192 D"},"headline":"A173018 \u2192 D","chain":null}},{"id":"l-889dbdaf23fbda76","kind":"fingerprint_related","node_a_id":"n-a06ada19a57d4cd7","node_b_id":"n-6e6daac86c9463fd","fingerprint":"f6-b00b790aa67df404","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b00b790aa67df404"},"other_id":"n-a06ada19a57d4cd7","other":{"id":"n-a06ada19a57d4cd7","name":"derivative of Triangle read by rows: T(n, k) = t(n-k, k), where t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), and f(n) = 3*n + 3.","first_index":0,"fingerprint":"f6-b00b790aa67df404","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["3"],["36","18"]],"card_rows":[["3"],["36","18"],["297","594","81"],["2106","10692","6318","324"]],"row_sums_preview":["0","3","54","972","19440","437400","11022480","308629440"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-dfedc6a36c97eba4","steps":["D"],"edges":["e-f8f0aa242fcc6513"],"text":"A257620 \u2192 D"},"headline":"A257620 \u2192 D","chain":null}},{"id":"l-a184aabc18f9f601","kind":"fingerprint_related","node_a_id":"n-1e9cff7ed448d4e0","node_b_id":"n-6e6daac86c9463fd","fingerprint":"f6-b00b790aa67df404","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b00b790aa67df404"},"other_id":"n-1e9cff7ed448d4e0","other":{"id":"n-1e9cff7ed448d4e0","name":"Coefficients of the derivatives of the Eulerian polynomials (with indexing as in A173018).","first_index":1,"fingerprint":"f6-b00b790aa67df404","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["11","22","3"],["26","132","78","4"],["57","604","906","228","5"]],"card_rows":[["11","22","3"],["26","132","78","4"],["57","604","906","228","5"],["120","2382","7248","4764","600","6"]],"row_sums_preview":["36","240","1800","15120","141120","1451520","16329600","199584000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-41809911e05bb4ba","steps":["D (shift +2)"],"edges":["e-2ca40a069e1a39d6"],"text":"A257609 \u2192 D (shift +2)"},"headline":"A257609 \u2192 D (shift +2)","chain":null}},{"id":"l-dfe6690172f5fe21","kind":"fingerprint_related","node_a_id":"n-b59d7fbabc61ed1c","node_b_id":"n-6e6daac86c9463fd","fingerprint":"f6-b00b790aa67df404","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b00b790aa67df404"},"other_id":"n-b59d7fbabc61ed1c","other":{"id":"n-b59d7fbabc61ed1c","name":"derivative of reversal of Triangle of Eulerian numbers T(n,k), 0 <= k <= n, read by rows.","first_index":0,"fingerprint":"f6-b00b790aa67df404","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],[],["1"]],"card_rows":[["1"],["4","2"],["11","22","3"],["26","132","78","4"]],"row_sums_preview":["0","0","1","6","36","240","1800","15120"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-efd1fdf07e5468d8","steps":["reverse (shift +1)","reverse","D"],"edges":["e-61bc78adfe1eb25a","e-52ca2f2e23126004","e-8f2aa23d1a9bfd26"],"text":"A173018 \u2192 reverse (shift +1) \u2192 reverse \u2192 D"},"headline":"A173018 \u2192 reverse (shift +1) \u2192 reverse \u2192 D","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:  3,    22,       11;\n   2:  4,    78,      132,        26;\n   3:  5,   228,      906,       604,         57;\n   4:  6,   600,     4764,      7248,       2382,         120;\n   5:  7,  1482,    21465,     62476,      46857,        8586,         247;\n   6:  8,  3514,    87648,    441170,     624760,      264702,       29216,        502;\n   7:  9,  8104,   334880,   2731152,    6551770,     5241416,     1365576,      95680,       1013;\n   8: 10, 18324,  1221096,  15424416,   58428684,    78621240,    38952456,    6610464,     305274,      2036;\n   9: 11, 40830,  4304439,  81501480,  464229318,   975073716,   812561430,  265273896,   30563055,    956542,    4083;\n  10: 12, 89958, 14797260, 409801050, 3386252280, 10539350556, 13651032024, 7528107540, 1693126140, 136600350, 2959452, 8178;\n","recurrence_sentence":"The row polynomials satisfy (n+1)*P(n,x) = -(n^2+3*n+2+(n^3+3*n^2+2*n)*x+(n^2+3*n+2)*x^2)*P(n-2,x) - ((n^2+3*n+2)*x-(n^2+2*n)*x^2-(n+2)*x^3)*d/dx P(n-2,x) + (n^2+3*n+2+(n^2+4*n+4)*x)*P(n-1,x) + ((n+1)*x-(n+1)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 3 + 22*x + 11*x^2, P(2,x) = 4 + 78*x + 132*x^2 + 26*x^3.","mathematica":"P[1, x_] := 3 + 22*x + 11*x^2;\nP[2, x_] := 4 + 78*x + 132*x^2 + 26*x^3;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^2 + 3*n + 2 + (n^3 + 3*n^2 + 2*n)*x + (n^2 + 3*n + 2)*x^2)*P[n - 2, x] - ((n^2 + 3*n + 2)*x - (n^2 + 2*n)*x^2 - (n + 2)*x^3)*D[P[n - 2, x], x] + (n^2 + 3*n + 2 + (n^2 + 4*n + 4)*x)*P[n - 1, x] + ((n + 1)*x - (n + 1)*x^2)*D[P[n - 1, x], x])/(n + 1)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 3 + 22*x + 11*x^2; v[2] = 4 + 78*x + 132*x^2 + 26*x^3; for(n = 3, N, my(i = n); v[i] = (-(n^2 + 3*n + 2 + (n^3 + 3*n^2 + 2*n)*x + (n^2 + 3*n + 2)*x^2)*v[i - 2] - ((n^2 + 3*n + 2)*x - (n^2 + 2*n)*x^2 - (n + 2)*x^3)*deriv(v[i - 2], x) + (n^2 + 3*n + 2 + (n^2 + 4*n + 4)*x)*v[i - 1] + ((n + 1)*x - (n + 1)*x^2)*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, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-41809911e05bb4ba","steps":["D (shift +2)","reverse"],"edges":["e-2ca40a069e1a39d6","e-0d05227aa568e6cc"],"text":"A257609 \u2192 D (shift +2) \u2192 reverse"},"headline":"A257609 \u2192 D (shift +2) \u2192 reverse","chain":null}
