{"id":"n-1b6003327f97748a","revision":0,"first_index":1,"fingerprint":"f6-90c2a235bbf6b62e","name":"reversal of one_plus_derivative of Triangle read by rows: T(n, k) = (k+1)*S2(n+1, k+1), for n >= k >= 0, and S2 = A048993 (Stirling2).","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":[["1"],["2","3"],["3","12","7"],["4","30","50","15"],["5","60","195","180","31"],["6","105","560","1050","602","63"],["7","168","1330","4200","5103","1932","127"],["8","252","2772","13230","27804","23310","6050","255"],["9","360","5250","35280","114135","170100","102315","18660","511"],["10","495","9240","83160","383922","897435","986920","437250","57002","1023"],["11","660","15345","178200","1113189","3764376","6618260","5517600","1834503","173052","2047"],["12","858","24310","353925","2876016","13297284","34292544","46606560","30034004","7597590","523250","4095"],["13","1092","37037","660660","6774768","41081040","146386240","295975680","317181865","160300140","31175235","1577940","8191"],["14","1365","54600","1171170","14794780","113963850","537027920","1516394880","2452447998","2103466365","843067680","127067850","4750202","16383"],["15","1680","78260","1987440","30328298","289369080","1743794910","6566274000","14992348371","19691295624","13674632790","4384762200","515396703","14283372","32767"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","5","22","99","471","2386","12867","73681","446620","2856457","19217243","135610448","1001159901","7714225057","61904585510"],"nonn":true,"definitions":[{"position":0,"source":"derived:reversal","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"],["2","3"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-1","1","1"],["0","-1","-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0","1","1"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["2","1"],["0","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","-1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":179187,"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-6f3d64abfa69a50f"]},{"id":179188,"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-6f3d64abfa69a50f"]},{"id":179189,"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-6f3d64abfa69a50f"]}],"region_exact":{},"incoming_edges":[{"id":"e-6f3d64abfa69a50f","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"reversal\",\"schedule\":{\"first_index\":1,\"first_degree\":0,\"step\":1}}"},"operation_detail":"reversal, shift 0","conclusion_id":"n-1b6003327f97748a","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":"e4671d716a5c788a96c934d7adc31555f509221770324e90fa4ff36bc34efe23","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-8be0b20fc2f56554","role":"sequence","name":"one_plus_derivative of Triangle read by rows: T(n, k) = (k+1)*S2(n+1, k+1), for n >= k >= 0, and S2 = A048993 (Stirling2).","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-08a074ffe5c333db","kind":"fingerprint_related","node_a_id":"n-8be0b20fc2f56554","node_b_id":"n-1b6003327f97748a","fingerprint":"f6-90c2a235bbf6b62e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-90c2a235bbf6b62e"},"other_id":"n-8be0b20fc2f56554","other":{"id":"n-8be0b20fc2f56554","name":"one_plus_derivative of Triangle read by rows: T(n, k) = (k+1)*S2(n+1, k+1), for n >= k >= 0, and S2 = A048993 (Stirling2).","first_index":1,"fingerprint":"f6-90c2a235bbf6b62e","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["3","2"],["7","12","3"]],"card_rows":[["1"],["3","2"],["7","12","3"],["15","50","30","4"]],"row_sums_preview":["1","5","22","99","471","2386","12867","73681"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3cb22164e2f83fe1","steps":["\u03b8+1","D+1"],"edges":["e-907e3c6c6275331d","e-8b35b367ffdd04e7"],"text":"A008277 \u2192 \u03b8+1 \u2192 D+1"},"headline":"A008277 \u2192 \u03b8+1 \u2192 D+1","chain":null}},{"id":"l-281255256038ea47","kind":"fingerprint_related","node_a_id":"n-d99a5c662672737d","node_b_id":"n-1b6003327f97748a","fingerprint":"f6-90c2a235bbf6b62e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-90c2a235bbf6b62e"},"other_id":"n-d99a5c662672737d","other":{"id":"n-d99a5c662672737d","name":"derivative of Triangle of Stirling numbers of the second kind, S2(n,k), n >= 1, 1 <= k <= n.","first_index":1,"fingerprint":"f6-90c2a235bbf6b62e","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["1"],["3","2"]],"card_rows":[["1"],["3","2"],["7","12","3"],["15","50","30","4"]],"row_sums_preview":["0","1","5","22","99","471","2386","12867"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3cb22164e2f83fe1","steps":["D"],"edges":["e-30216415fa236e17"],"text":"A008277 \u2192 D"},"headline":"A008277 \u2192 D","chain":null}},{"id":"l-5e8e0800ebd59fbf","kind":"fingerprint_related","node_a_id":"n-8d7a43e42ddb80a0","node_b_id":"n-1b6003327f97748a","fingerprint":"f6-90c2a235bbf6b62e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-90c2a235bbf6b62e"},"other_id":"n-8d7a43e42ddb80a0","other":{"id":"n-8d7a43e42ddb80a0","name":"derivative of basis_map_A008277 of a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.","first_index":0,"fingerprint":"f6-90c2a235bbf6b62e","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["1"],["3","2"]],"card_rows":[["1"],["3","2"],["7","12","3"],["15","50","30","4"]],"row_sums_preview":["0","1","5","22","99","471","2386","12867"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3cb22164e2f83fe1","steps":["\u03b8+1","D+1"],"edges":["e-41f88f689ebf91d9","e-9ee43c12e3aef4ff"],"text":"A008277 \u2192 \u03b8+1 \u2192 D+1"},"headline":"A008277 \u2192 \u03b8+1 \u2192 D+1","chain":null}},{"id":"l-cca752b33e47d0f2","kind":"fingerprint_related","node_a_id":"n-1b6003327f97748a","node_b_id":"n-9c06e18062eecdb1","fingerprint":"f6-90c2a235bbf6b62e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-90c2a235bbf6b62e"},"other_id":"n-9c06e18062eecdb1","other":{"id":"n-9c06e18062eecdb1","name":"derivative of basis_map_A008277 of basis-map coefficients of Triangle of coefficients in expansion of x^n in terms of Laguerre polynomials L_n(x).","first_index":1,"fingerprint":"f6-90c2a235bbf6b62e","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":0,"preview_rows":[[],["-1"],["6","4"]],"card_rows":[["-1"],["6","4"],["-42","-72","-18"],["360","1200","720","96"]],"row_sums_preview":["0","-1","10","-132","2376","-56520","1717920","-64849680"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-cf1765de6ac2bd4b","steps":["basis A008277","D"],"edges":["e-2b559797932c7395","e-b97d25183820371d"],"text":"basis-map coefficients of A021012 \u2192 basis A008277 \u2192 D"},"headline":"basis-map coefficients of A021012 \u2192 basis A008277 \u2192 D","chain":null}},{"id":"l-e5e3811ddfda7a19","kind":"fingerprint_related","node_a_id":"n-1b6003327f97748a","node_b_id":"n-1dcf621c8ad470a0","fingerprint":"f6-90c2a235bbf6b62e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-90c2a235bbf6b62e"},"other_id":"n-1dcf621c8ad470a0","other":{"id":"n-1dcf621c8ad470a0","name":"derivative of basis_map_A008277 of Infinite lower triangular matrix with 2^k in the right diagonal and the rest zeros. Triangle, T(n,k), n zeros followed by the term 2^k. Triangle by columns, (2^k, 0, 0, 0, ...).","first_index":0,"fingerprint":"f6-90c2a235bbf6b62e","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["2"],["12","8"]],"card_rows":[["2"],["12","8"],["56","96","24"],["240","800","480","64"]],"row_sums_preview":["0","2","20","176","1584","15072","152704","1646976"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ae7910bbe46f8a41","steps":["basis A008277","D"],"edges":["e-dbb6853ea73a13a7","e-0897e632739046bb"],"text":"A130123 \u2192 basis A008277 \u2192 D"},"headline":"A130123 \u2192 basis A008277 \u2192 D","chain":null}},{"id":"l-ecaee4e040dcb9ec","kind":"fingerprint_related","node_a_id":"n-b22d1ac5cb7f71d6","node_b_id":"n-1b6003327f97748a","fingerprint":"f6-90c2a235bbf6b62e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-90c2a235bbf6b62e"},"other_id":"n-b22d1ac5cb7f71d6","other":{"id":"n-b22d1ac5cb7f71d6","name":"derivative of basis_map_A008277 of basis-map coefficients of A scaled Hermite triangle.","first_index":1,"fingerprint":"f6-90c2a235bbf6b62e","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["3","2"],["7","12","3"],["15","50","30","4"]],"card_rows":[["3","2"],["7","12","3"],["15","50","30","4"],["31","180","195","60","5"]],"row_sums_preview":["5","22","99","471","2386","12867","73681","446620"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-5fe240122e98ffb3","steps":["basis A008277","D"],"edges":["e-66762082773dd3e0","e-d6d7127dbccbcb15"],"text":"basis-map coefficients of A112227 \u2192 basis A008277 \u2192 D"},"headline":"basis-map coefficients of A112227 \u2192 basis A008277 \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:  1;\n   2:  2,   3;\n   3:  3,  12,    7;\n   4:  4,  30,   50,    15;\n   5:  5,  60,  195,   180,     31;\n   6:  6, 105,  560,  1050,    602,     63;\n   7:  7, 168, 1330,  4200,   5103,   1932,    127;\n   8:  8, 252, 2772, 13230,  27804,  23310,   6050,    255;\n   9:  9, 360, 5250, 35280, 114135, 170100, 102315,  18660,   511;\n  10: 10, 495, 9240, 83160, 383922, 897435, 986920, 437250, 57002, 1023;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(1+(n-1)*x+(n-1)*x^2)*P(n-2,x) + (x^2+x^3)*d/dx P(n-2,x) + (2+(n+1)*x)*P(n-1,x) - x^2*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2 + 3*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 2 + 3*x;\nP[n_, x_] := P[n, x] = Expand[-(1 + (n - 1)*x + (n - 1)*x^2)*P[n - 2, x] + (x^2 + x^3)*D[P[n - 2, x], x] + (2 + (n + 1)*x)*P[n - 1, x] - 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; v[2] = 2 + 3*x; for(n = 3, N, my(i = n); v[i] = -(1 + (n - 1)*x + (n - 1)*x^2)*v[i - 2] + (x^2 + x^3)*deriv(v[i - 2], x) + (2 + (n + 1)*x)*v[i - 1] - 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":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-3cb22164e2f83fe1","steps":["\u03b8+1","D+1","reverse"],"edges":["e-907e3c6c6275331d","e-8b35b367ffdd04e7","e-6f3d64abfa69a50f"],"text":"A008277 \u2192 \u03b8+1 \u2192 D+1 \u2192 reverse"},"headline":"A008277 \u2192 \u03b8+1 \u2192 D+1 \u2192 reverse","chain":null}