{"id":"n-fabe4ac480a936a7","revision":0,"first_index":0,"fingerprint":"f6-5cd009bfc7bc0bb1","name":"shift_x_plus_one of derivative of Triangle read by rows, T(n, k) = binomial(n, k)*binomial(n + 2, k).","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"],["20","12"],["105","120","30"],["504","840","420","60"],["2310","5040","3780","1120","105"],["10296","27720","27720","12600","2520","168"],["45045","144144","180180","110880","34650","5040","252"],["194480","720720","1081080","840840","360360","83160","9240","360"],["831402","3500640","6126120","5765760","3153150","1009008","180180","15840","495"],["3527160","16628040","33256080","36756720","24504480","10090080","2522520","360360","25740","660"],["14872858","77597520","174594420","221707200","174594420","88216128","28588560","5765760","675675","40040","858"],["62403600","356948592","892371480","1280359080","1163962800","698377680","279351072","73513440","12252240","1201200","60060","1092"],["260757900","1622493600","4461857400","7138971840","7362064710","5121436320","2444321880","798145920","174594420","24504480","2042040","87360","1365"],["1085822640","7301221200","21903663600","38669430800","44618574000","35337910608","19632172560","7682154480","2095133040","387987600","46558512","3341520","123760","1680"]],"degrees":[-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["0","3","32","255","1824","12355","81024","520191","3290240","20582595","127671840","786653439","4820802336","29411279235","178764096000"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["3"]],"recurrence":{"denominator":{"coeffs":[["-1"],["0"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","0"],["2","0"],["4","3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-1","-1"],["0","-2","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":177557,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, -1]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, -1]","region_exact":null,"edges":["e-186b8efe43b74013"]},{"id":177558,"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-186b8efe43b74013","e-e2cb16f67cf9b10f"]},{"id":177559,"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-186b8efe43b74013","e-e2cb16f67cf9b10f"]},{"id":177560,"position":3,"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-e2cb16f67cf9b10f"]}],"region_exact":{},"incoming_edges":[{"id":"e-186b8efe43b74013","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-fabe4ac480a936a7","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":"50022666adb6e9725950be3902bc75ebe005a0cc17dc1353f7ee8e940c54f05c","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}}],"premises":[{"position":0,"node_id":"n-82cc9c378f27e13e","role":"sequence","name":"derivative of Triangle read by rows, T(n, k) = binomial(n, k)*binomial(n + 2, k).","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]"}]},{"id":"e-e2cb16f67cf9b10f","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-fabe4ac480a936a7","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":"ccbe4df39fa702a11c1a32711710cb0c0e3ed360451b66eea80086ea05b2d2f9","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-5aa52e3acf29a627","role":"sequence","name":"Triangle read by rows, related to clusters of type D.","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-0ef08017187eb388","kind":"fingerprint_related","node_a_id":"n-fabe4ac480a936a7","node_b_id":"n-3a81c23f7a892ffd","fingerprint":"f6-5cd009bfc7bc0bb1","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-5cd009bfc7bc0bb1"},"other_id":"n-3a81c23f7a892ffd","other":{"id":"n-3a81c23f7a892ffd","name":"mobius_x_over_x_plus_one of reversal of Triangle read by rows: T(n,k) = number of vertices of degree k in an origami flip graph OFG(A2n).","first_index":0,"fingerprint":"f6-5cd009bfc7bc0bb1","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["8"],["18","30"],["32","128","112"]],"card_rows":[["8"],["18","30"],["32","128","112"],["50","350","700","420"]],"row_sums_preview":["8","48","272","1520","8472","47264","264224","1480608"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7a4827a398e28fe4","steps":["reverse","x \u21a6 x/(x+1)"],"edges":["e-8b10d4d15ed56acc","e-14c2f28924cda6a6"],"text":"A352880 \u2192 reverse \u2192 x \u21a6 x/(x+1)"},"headline":"A352880 \u2192 reverse \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-4c43907ae7a33e56","kind":"fingerprint_related","node_a_id":"n-8cac8e8480894b25","node_b_id":"n-fabe4ac480a936a7","fingerprint":"f6-5cd009bfc7bc0bb1","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-5cd009bfc7bc0bb1"},"other_id":"n-8cac8e8480894b25","other":{"id":"n-8cac8e8480894b25","name":"shift_x_plus_one of reversal of Triangle of Dyck paths counted by number of long interior inclines.","first_index":0,"fingerprint":"f6-5cd009bfc7bc0bb1","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["2"],["5","3"],["14","16","4"]],"card_rows":[["2"],["5","3"],["14","16","4"],["42","70","35","5"]],"row_sums_preview":["2","8","34","152","706","3376","16514","82256"],"regions":["(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c963abfc0266d66c","steps":["reverse","x \u21a6 x+1"],"edges":["e-7ac651410315272d","e-ca102fc79ed966ef"],"text":"A108838 \u2192 reverse \u2192 x \u21a6 x+1"},"headline":"A108838 \u2192 reverse \u2192 x \u21a6 x+1","chain":null}},{"id":"l-5b89d544ff453b5f","kind":"fingerprint_related","node_a_id":"n-f7cfaa6a3770b14b","node_b_id":"n-fabe4ac480a936a7","fingerprint":"f6-5cd009bfc7bc0bb1","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-5cd009bfc7bc0bb1"},"other_id":"n-f7cfaa6a3770b14b","other":{"id":"n-f7cfaa6a3770b14b","name":"mobius_x_over_x_plus_one of Triangle of Dyck paths counted by number of long interior inclines.","first_index":0,"fingerprint":"f6-5cd009bfc7bc0bb1","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["2"],["3","5"],["4","16","14"]],"card_rows":[["2"],["3","5"],["4","16","14"],["5","35","70","42"]],"row_sums_preview":["2","8","34","152","706","3376","16514","82256"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c963abfc0266d66c","steps":["x \u21a6 x/(x+1)"],"edges":["e-c9da786527cfa7e3"],"text":"A108838 \u2192 x \u21a6 x/(x+1)"},"headline":"A108838 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-ee59c83cf6254f8c","kind":"fingerprint_related","node_a_id":"n-d54c958727b939eb","node_b_id":"n-fabe4ac480a936a7","fingerprint":"f6-5cd009bfc7bc0bb1","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-5cd009bfc7bc0bb1"},"other_id":"n-d54c958727b939eb","other":{"id":"n-d54c958727b939eb","name":"shift_x_plus_one of Triangle read by rows: T(n,k) = number of vertices of degree k in an origami flip graph OFG(A2n).","first_index":0,"fingerprint":"f6-5cd009bfc7bc0bb1","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["8"],["30","18"],["112","128","32"]],"card_rows":[["8"],["30","18"],["112","128","32"],["420","700","350","50"]],"row_sums_preview":["8","48","272","1520","8472","47264","264224","1480608"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7a4827a398e28fe4","steps":["x \u21a6 x+1"],"edges":["e-5e1fa9e41a7375d8"],"text":"A352880 \u2192 x \u21a6 x+1"},"headline":"A352880 \u2192 x \u21a6 x+1","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:      3;\n  2:     20,      12;\n  3:    105,     120,      30;\n  4:    504,     840,     420,      60;\n  5:   2310,    5040,    3780,    1120,     105;\n  6:  10296,   27720,   27720,   12600,    2520,     168;\n  7:  45045,  144144,  180180,  110880,   34650,    5040,    252;\n  8: 194480,  720720, 1081080,  840840,  360360,   83160,   9240,   360;\n  9: 831402, 3500640, 6126120, 5765760, 3153150, 1009008, 180180, 15840, 495;\n","recurrence_sentence":"The row polynomials satisfy (n^2-1)*P(n,x) = (4*n^2+2*n+3*n^2*x)*P(n-1,x) - ((2*n+1)*x+(2*n+1)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 3.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 3;\nP[n_, x_] := P[n, x] = Expand[Together[((4*n^2 + 2*n + 3*n^2*x)*P[n - 1, x] - ((2*n + 1)*x + (2*n + 1)*x^2)*D[P[n - 1, x], x])/(n^2 - 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 3; for(n = 2, N, my(i = n + 1); v[i] = ((4*n^2 + 2*n + 3*n^2*x)*v[i - 1] - ((2*n + 1)*x + (2*n + 1)*x^2)*deriv(v[i - 1], x))/(n^2 - 1)); 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-5aa52e3acf29a627","steps":["D"],"edges":["e-e2cb16f67cf9b10f"],"text":"A345013 \u2192 D"},"headline":"A345013 \u2192 D","chain":null}