{"id":"n-566cddcc2128607a","revision":0,"first_index":1,"fingerprint":"f6-867bd83b37df747f","name":"shift_x_plus_one of derivative of Euler's triangle: triangle of Eulerian numbers T(n,k) (n>=0, 0 <= k <= n) read by rows.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-3,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[[],[],["1"],["6","2"],["36","28","3"],["240","300","90","4"],["1800","3120","1620","248","5"],["15120","33600","25200","7224","630","6"],["141120","383040","378000","163296","28980","1524","7"],["1451520","4656960","5715360","3336480","932400","108900","3570","8"],["16329600","60480000","88905600","65741760","25515000","4911120","391860","8176","9"],["199584000","838252800","1437004800","1289977920","646153200","177645600","24486000","1368048","18414","10"],["2634508800","12374208000","24249456000","25647874560","15810379200","5718176640","1158696000","117408192","4672404","40940","11"],["37362124800","194075481600","428107680000","521824343040","382961779200","172834421760","46979412480","7208160960","547026480","15697500","90090","12"],["566658892800","3225596774400","7911429926400","10926345830400","9317179872000","5059108454400","1740336998400","365393508480","43281037800","2494018800","52072020","196584","13"]],"degrees":[-1,-1,0,1,2,3,4,5,6,7,8,9,10,11,12],"row_sums":["0","0","1","8","67","634","6793","81780","1095967","16205198","262283125","4614490792","87715420747","1791916217922","39157877582497"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["0"],["0"],["1"]],"recurrence":{"denominator":{"coeffs":[["-3"],["1"]]},"terms":[{"coeff":{"coeffs":[["12","32","24"],["-16","-40","-26"],["7","16","9"],["-1","-2","-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","4","10","6"],["0","-4","-9","-5"],["0","1","2","1"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["10","18"],["-9","-12"],["2","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","3","3"],["0","-1","-1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":176787,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, -1]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"-1"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, -1]","region_exact":null,"edges":["e-e4db67e5c16b49a7"]},{"id":176788,"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-86a8b73a80030b16","e-e4db67e5c16b49a7"]},{"id":176789,"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-86a8b73a80030b16","e-e4db67e5c16b49a7"]},{"id":176790,"position":3,"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-86a8b73a80030b16"]}],"region_exact":{},"incoming_edges":[{"id":"e-86a8b73a80030b16","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-566cddcc2128607a","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":"76d8e0267e4b2f97978019c73361d2538b86fa8386f2a3b16cdacbf6b8197bb2","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-d6d713aa669b74c1","role":"sequence","name":"shift_x_plus_one of Euler's triangle: triangle of Eulerian numbers T(n,k) (n>=0, 0 <= k <= n) read by rows.","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]"}]},{"id":"e-e4db67e5c16b49a7","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-566cddcc2128607a","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":null,"lean_declarations":["Polynomial.Splits.comp_of_degree_le_one"],"certificate_hash":"0ea0c069ac6ca8ef9e8bc5b0ab8a3e4dd5c53fc57aac998d17a292dbf65d50cf","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"-1"}}}],"premises":[{"position":0,"node_id":"n-08acbfdfd91c4faa","role":"sequence","name":"derivative of Euler's triangle: triangle of Eulerian numbers T(n,k) (n>=0, 0 <= k <= n) read by rows.","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-7e01acd42fdde3fb","kind":"fingerprint_related","node_a_id":"n-566cddcc2128607a","node_b_id":"n-2422dc9bfda6e079","fingerprint":"f6-867bd83b37df747f","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-867bd83b37df747f"},"other_id":"n-2422dc9bfda6e079","other":{"id":"n-2422dc9bfda6e079","name":"shift_x_plus_one of Coefficients of the derivatives of the Eulerian polynomials (with indexing as in A173018).","first_index":1,"fingerprint":"f6-867bd83b37df747f","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["36","28","3"],["240","300","90","4"],["1800","3120","1620","248","5"]],"card_rows":[["36","28","3"],["240","300","90","4"],["1800","3120","1620","248","5"],["15120","33600","25200","7224","630","6"]],"row_sums_preview":["67","634","6793","81780","1095967","16205198","262283125","4614490792"],"regions":["(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-41809911e05bb4ba","steps":["D (shift +2)","x \u21a6 x+1"],"edges":["e-2ca40a069e1a39d6","e-f3345bf0827a6cc1"],"text":"A257609 \u2192 D (shift +2) \u2192 x \u21a6 x+1"},"headline":"A257609 \u2192 D (shift +2) \u2192 x \u21a6 x+1","chain":null}},{"id":"l-add306e6eb5df0b0","kind":"fingerprint_related","node_a_id":"n-566cddcc2128607a","node_b_id":"n-b8d0962eb2908714","fingerprint":"f6-867bd83b37df747f","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-867bd83b37df747f"},"other_id":"n-b8d0962eb2908714","other":{"id":"n-b8d0962eb2908714","name":"derivative of T(n, k) = Sum_{j=0..n-k} (-1)^j*binomial(n - k + 1, j)*(n - k + 1 - j)^n. Triangle read by rows, T(n, k) for 1 <= k <= n.","first_index":0,"fingerprint":"f6-867bd83b37df747f","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["1"],["6","2"]],"card_rows":[["1"],["6","2"],["36","28","3"],["240","300","90","4"]],"row_sums_preview":["0","1","8","67","634","6793","81780","1095967"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3b2b9e31bd278a34","steps":["D"],"edges":["e-c1e8d829aae6c4c6"],"text":"A090582 \u2192 D"},"headline":"A090582 \u2192 D","chain":null}},{"id":"l-ee6ecd400e8b06d3","kind":"fingerprint_related","node_a_id":"n-566cddcc2128607a","node_b_id":"n-c2d241782f7374a5","fingerprint":"f6-867bd83b37df747f","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-867bd83b37df747f"},"other_id":"n-c2d241782f7374a5","other":{"id":"n-c2d241782f7374a5","name":"shift_x_plus_one of derivative of eulerian","first_index":1,"fingerprint":"f6-867bd83b37df747f","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["1"],["6","2"]],"card_rows":[["1"],["6","2"],["36","28","3"],["240","300","90","4"]],"row_sums_preview":["0","1","8","67","634","6793","81780","1095967"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ff270bcb802a85af","steps":["D","x \u21a6 x+1"],"edges":["e-0215d14e0b41ed17","e-77700992f4cf9e45"],"text":"A008292 \u2192 D \u2192 x \u21a6 x+1"},"headline":"A008292 \u2192 D \u2192 x \u21a6 x+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:       0;\n   2:       0;\n   3:       1;\n   4:       6,       2;\n   5:      36,      28,       3;\n   6:     240,     300,      90,       4;\n   7:    1800,    3120,    1620,     248,      5;\n   8:   15120,   33600,   25200,    7224,    630,      6;\n   9:  141120,  383040,  378000,  163296,  28980,   1524,    7;\n  10: 1451520, 4656960, 5715360, 3336480, 932400, 108900, 3570, 8;\n","recurrence_sentence":"The row polynomials satisfy (n-3)*P(n,x) = -(n^3-7*n^2+16*n-12+(2*n^3-16*n^2+40*n-32)*x+(n^3-9*n^2+26*n-24)*x^2)*P(n-2,x) + ((n^2-4*n+4)*x+(2*n^2-9*n+10)*x^2+(n^2-5*n+6)*x^3)*d/dx P(n-2,x) + (2*n^2-9*n+10+(2*n^2-12*n+18)*x)*P(n-1,x) - ((n-3)*x+(n-3)*x^2)*d/dx P(n-1,x) for n >= 4, with P(1,x) = 0, P(2,x) = 0, P(3,x) = 1.","mathematica":"P[1, x_] := 0;\nP[2, x_] := 0;\nP[3, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^3 - 7*n^2 + 16*n - 12 + (2*n^3 - 16*n^2 + 40*n - 32)*x + (n^3 - 9*n^2 + 26*n - 24)*x^2)*P[n - 2, x] + ((n^2 - 4*n + 4)*x + (2*n^2 - 9*n + 10)*x^2 + (n^2 - 5*n + 6)*x^3)*D[P[n - 2, x], x] + (2*n^2 - 9*n + 10 + (2*n^2 - 12*n + 18)*x)*P[n - 1, x] - ((n - 3)*x + (n - 3)*x^2)*D[P[n - 1, x], x])/(n - 3)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 0; v[3] = 1; for(n = 4, N, my(i = n); v[i] = (-(n^3 - 7*n^2 + 16*n - 12 + (2*n^3 - 16*n^2 + 40*n - 32)*x + (n^3 - 9*n^2 + 26*n - 24)*x^2)*v[i - 2] + ((n^2 - 4*n + 4)*x + (2*n^2 - 9*n + 10)*x^2 + (n^2 - 5*n + 6)*x^3)*deriv(v[i - 2], x) + (2*n^2 - 9*n + 10 + (2*n^2 - 12*n + 18)*x)*v[i - 1] - ((n - 3)*x + (n - 3)*x^2)*deriv(v[i - 1], x))/(n - 3)); 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-efd1fdf07e5468d8","steps":["x \u21a6 x+1","D"],"edges":["e-aed75c9f11c92985","e-86a8b73a80030b16"],"text":"A173018 \u2192 x \u21a6 x+1 \u2192 D"},"headline":"A173018 \u2192 x \u21a6 x+1 \u2192 D","chain":null}
