{"id":"n-cbbf33139cbba3a7","revision":1,"first_index":1,"fingerprint":"f6-4732d0911d5a78c5","name":"basis_map_A008277 of basis-map coefficients of Triangle of coefficients in expansion of x^n in terms of Laguerre polynomials L_n(x).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":false,"rows":[["1"],["-1","-1"],["2","6","2"],["-6","-42","-36","-6"],["24","360","600","240","24"],["-120","-3720","-10800","-7800","-1800","-120"],["720","45360","216720","252000","100800","15120","720"],["-5040","-640080","-4868640","-8573040","-5292000","-1340640","-141120","-5040"],["40320","10281600","121968000","313286400","280264320","106686720","18627840","1451520","40320"],["-362880","-185431680","-3385670400","-12376022400","-15431472000","-8283461760","-2133734400","-272160000","-16329600","-362880"],["3628800","3712262400","103424428800","528897600000","895333824000","651322425600","232196025600","43110144000","4191264000","199584000","3628800"],["-39916800","-81709689600","-3453841036800","-24409163116800","-55061233920000","-52835952153600","-25043640652800","-6347848953600","-889146720000","-68058144000","-2634508800","-39916800"],["479001600","1961511552000","125318793600000","1213085922048000","3596583992601600","4464923362099200","2737697240678400","909917187379200","172202033203200","18836737920000","1164452889600","37362124800","479001600"],["-6227020800","-51005527372800","-4912932600576000","-64709612263296000","-249548076505728000","-395019614147558400","-307174452609024000","-130221451616256000","-31976561320704000","-4687402361241600","-411394356172800","-20966379033600","-566658892800","-6227020800"],["87178291200","1428241944729600","207057246607411200","3692519343152640000","18374299927087104000","36675320659475097600","35633370953750169600","18885244917547008000","5852147049036288000","1103908189063680000","128978363907993600","9281872664064000","396661224960000","9153720576000","87178291200"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","-2","10","-90","1248","-24360","631440","-20865600","852647040","-42085008000","2462394816000","-168193308729600","13241729554099200","-1188734048799897600","120563962753538304000"],"nonn":false,"definitions":[{"position":0,"source":"derived:basis_map_A008277","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","1"],["-1","-1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1"],["0","-1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":162408,"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-f4658f48c67ff3dd"]},{"id":162409,"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-f4658f48c67ff3dd"]},{"id":162410,"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-f4658f48c67ff3dd"]}],"region_exact":{},"incoming_edges":[{"id":"e-f4658f48c67ff3dd","operation":{"map":"A008277","operation":"basis_map"},"operation_detail":"A008277","conclusion_id":"n-cbbf33139cbba3a7","premise_count":1,"theorem":"ray basis maps: for L f = (a + b t) f + kappa (t - h) f', b > 0, kappa >= 0, and g real-rooted with every root <= h, of multiplicity m at h, T(u^k) = L^k g sends every real polynomial with all roots in S to one with all roots in (-inf, h] if and only if S lies in (-inf, a + b h + kappa m] (RealRooted rayBasisMap_maps_realRootedIn_iff; the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"9a150c374484e0304b20e6295e720630f5f6a3272c1ff94d3c33d0d4fd3e4ac3","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-cf1765de6ac2bd4b","role":"sequence","name":"basis-map coefficients of Triangle of coefficients in expansion of x^n in terms of Laguerre polynomials L_n(x).","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-195046033fe5f65b","kind":"fingerprint_related","node_a_id":"n-fb3d943c590f1977","node_b_id":"n-cbbf33139cbba3a7","fingerprint":"f6-4732d0911d5a78c5","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4732d0911d5a78c5"},"other_id":"n-fb3d943c590f1977","other":{"id":"n-fb3d943c590f1977","name":"basis_map_A008277 of basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k). shifted by x -> -x + 0","first_index":0,"fingerprint":"f6-4732d0911d5a78c5","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":0,"preview_rows":[["1"],["-1","-1"],["1","3","1"]],"card_rows":[["1"],["-1","-1"],["1","3","1"],["-1","-7","-6","-1"]],"row_sums_preview":["1","-2","5","-15","52","-203","877","-4140"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-477d9807e1cbb785","steps":["basis A008277"],"edges":["e-c0011ff889c7ff47"],"text":"basis-map coefficients of A030528, shifted \u2192 basis A008277"},"headline":"basis-map coefficients of A030528, shifted \u2192 basis A008277","chain":null}},{"id":"l-1dff6a8cb83f696b","kind":"fingerprint_related","node_a_id":"n-cae59b180fd3bbb3","node_b_id":"n-cbbf33139cbba3a7","fingerprint":"f6-4732d0911d5a78c5","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4732d0911d5a78c5"},"other_id":"n-cae59b180fd3bbb3","other":{"id":"n-cae59b180fd3bbb3","name":"basis_map_A008277 of basis-map coefficients of A scaled Hermite triangle.","first_index":1,"fingerprint":"f6-4732d0911d5a78c5","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1","3","1"],["1","7","6","1"],["1","15","25","10","1"]],"card_rows":[["1","3","1"],["1","7","6","1"],["1","15","25","10","1"],["1","31","90","65","15","1"]],"row_sums_preview":["5","15","52","203","877","4140","21147","115975"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-5fe240122e98ffb3","steps":["basis A008277"],"edges":["e-66762082773dd3e0"],"text":"basis-map coefficients of A112227 \u2192 basis A008277"},"headline":"basis-map coefficients of A112227 \u2192 basis A008277","chain":null}},{"id":"l-2e8cfa24755f5614","kind":"fingerprint_related","node_a_id":"n-3cb22164e2f83fe1","node_b_id":"n-cbbf33139cbba3a7","fingerprint":"f6-4732d0911d5a78c5","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4732d0911d5a78c5"},"other_id":"n-3cb22164e2f83fe1","other":{"id":"n-3cb22164e2f83fe1","name":"Triangle of Stirling numbers of the second kind, S2(n,k), n >= 1, 1 <= k <= n.","first_index":1,"fingerprint":"f6-4732d0911d5a78c5","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","1"],["1","3","1"]],"card_rows":[["1"],["1","1"],["1","3","1"],["1","7","6","1"]],"row_sums_preview":["1","2","5","15","52","203","877","4140"],"regions":["(-inf, 0)","(-inf, 0]"],"region_exact":{},"oeis_ids":["A008277"],"derivation":{"root":"n-3cb22164e2f83fe1","steps":[],"edges":[],"text":"A008277"},"headline":"A008277: Triangle of Stirling numbers of the second kind, S2(n,k), n >= 1, 1 <= k <= n.","chain":null}},{"id":"l-3194278a55d175be","kind":"fingerprint_related","node_a_id":"n-f277d949034445ba","node_b_id":"n-cbbf33139cbba3a7","fingerprint":"f6-4732d0911d5a78c5","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4732d0911d5a78c5"},"other_id":"n-f277d949034445ba","other":{"id":"n-f277d949034445ba","name":"Reflected triangle of Stirling numbers of 2nd kind, S(n,n-k+1), n >= 1, 1 <= k <= n.","first_index":1,"fingerprint":"f6-4732d0911d5a78c5","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","1"],["1","3","1"]],"card_rows":[["1"],["1","1"],["1","3","1"],["1","6","7","1"]],"row_sums_preview":["1","2","5","15","52","203","877","4140"],"regions":["(-inf, 0)","(-inf, 0]"],"region_exact":{},"oeis_ids":["A008278"],"derivation":{"root":"n-f277d949034445ba","steps":[],"edges":[],"text":"A008278"},"headline":"A008278: Reflected triangle of Stirling numbers of 2nd kind, S(n,n-k+1), n >= 1, 1 <= k <= n.","chain":null}},{"id":"l-53b532917af377aa","kind":"fingerprint_related","node_a_id":"n-af8b556000f2c0de","node_b_id":"n-cbbf33139cbba3a7","fingerprint":"f6-4732d0911d5a78c5","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4732d0911d5a78c5"},"other_id":"n-af8b556000f2c0de","other":{"id":"n-af8b556000f2c0de","name":"basis_map_A008277 of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals. shifted by x -> -x + 0","first_index":1,"fingerprint":"f6-4732d0911d5a78c5","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":0,"preview_rows":[["1"],["-1","-1"],["1","3","1"]],"card_rows":[["1"],["-1","-1"],["1","3","1"],["-1","-7","-6","-1"]],"row_sums_preview":["1","-2","5","-15","52","-203","877","-4140"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3cc7875cf81a5a84","steps":["basis A008277"],"edges":["e-9c20f6f9d09a133d"],"text":"basis-map coefficients of A026729, shifted \u2192 basis A008277"},"headline":"basis-map coefficients of A026729, shifted \u2192 basis A008277","chain":null}},{"id":"l-62ee043b171dae49","kind":"fingerprint_related","node_a_id":"n-4ee613bf6b082e7d","node_b_id":"n-cbbf33139cbba3a7","fingerprint":"f6-4732d0911d5a78c5","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4732d0911d5a78c5"},"other_id":"n-4ee613bf6b082e7d","other":{"id":"n-4ee613bf6b082e7d","name":"Triangle of Stirling numbers of 2nd kind, S(n, n-k), n >= 0, 0 <= k <= n.","first_index":1,"fingerprint":"f6-4732d0911d5a78c5","polytool_family":null,"degree_pattern":"irregular","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1"],["1","1"]],"card_rows":[["1"],["1"],["1","1"],["1","3","1"]],"row_sums_preview":["1","1","2","5","15","52","203","877"],"regions":["(-inf, 0)"],"region_exact":{},"oeis_ids":["A106800"],"derivation":{"root":"n-4ee613bf6b082e7d","steps":[],"edges":[],"text":"A106800"},"headline":"A106800: Triangle of Stirling numbers of 2nd kind, S(n, n-k), n >= 0, 0 <= k <= n.","chain":null}},{"id":"l-869ce59aa3337700","kind":"fingerprint_related","node_a_id":"n-95ac664928981c03","node_b_id":"n-cbbf33139cbba3a7","fingerprint":"f6-4732d0911d5a78c5","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4732d0911d5a78c5"},"other_id":"n-95ac664928981c03","other":{"id":"n-95ac664928981c03","name":"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":"f6-4732d0911d5a78c5","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","2"],["4","12","4"]],"card_rows":[["1"],["2","2"],["4","12","4"],["8","56","48","8"]],"row_sums_preview":["1","4","20","120","832","6496","56128","529920"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ae7910bbe46f8a41","steps":["basis A008277"],"edges":["e-dbb6853ea73a13a7"],"text":"A130123 \u2192 basis A008277"},"headline":"A130123 \u2192 basis A008277","chain":null}},{"id":"l-9503c00cb2cdf46f","kind":"fingerprint_related","node_a_id":"n-e65f982fa1b436c5","node_b_id":"n-cbbf33139cbba3a7","fingerprint":"f6-4732d0911d5a78c5","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4732d0911d5a78c5"},"other_id":"n-e65f982fa1b436c5","other":{"id":"n-e65f982fa1b436c5","name":"Triangle of Stirling numbers of 2nd kind, S(n,k), n >= 0, 0 <= k <= n.","first_index":0,"fingerprint":"f6-4732d0911d5a78c5","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["0","1"],["0","1","1"]],"card_rows":[["1"],["0","1"],["0","1","1"],["0","1","3","1"]],"row_sums_preview":["1","1","2","5","15","52","203","877"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":["A048993"],"derivation":{"root":"n-4ee613bf6b082e7d","steps":["reverse (shift +1)"],"edges":["e-ad13a5d17ca116d8"],"text":"A106800 \u2192 reverse (shift +1)"},"headline":"A048993: Triangle of Stirling numbers of 2nd kind, S(n,k), n >= 0, 0 <= k <= n.","chain":"A106800 \u2192 reverse (shift +1)"}},{"id":"l-e196a253071e3f6e","kind":"fingerprint_related","node_a_id":"n-cbbf33139cbba3a7","node_b_id":"n-71012aa9360ef4f9","fingerprint":"f6-4732d0911d5a78c5","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4732d0911d5a78c5"},"other_id":"n-71012aa9360ef4f9","other":{"id":"n-71012aa9360ef4f9","name":"reversal of Triangle of Stirling numbers of 2nd kind, S(n,k), n >= 0, 0 <= k <= n.","first_index":0,"fingerprint":"f6-4732d0911d5a78c5","polytool_family":null,"degree_pattern":"irregular","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1"],["1","1"]],"card_rows":[["1"],["1"],["1","1"],["1","3","1"]],"row_sums_preview":["1","1","2","5","15","52","203","877"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-4ee613bf6b082e7d","steps":["reverse (shift +1)","reverse"],"edges":["e-ad13a5d17ca116d8","e-c300844494b58a7a"],"text":"A106800 \u2192 reverse (shift +1) \u2192 reverse"},"headline":"A106800 \u2192 reverse (shift +1) \u2192 reverse","chain":null}},{"id":"l-fb22d4355c76c215","kind":"fingerprint_related","node_a_id":"n-f3c2281829f25122","node_b_id":"n-cbbf33139cbba3a7","fingerprint":"f6-4732d0911d5a78c5","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4732d0911d5a78c5"},"other_id":"n-f3c2281829f25122","other":{"id":"n-f3c2281829f25122","name":"basis_map_A008277 of a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.","first_index":0,"fingerprint":"f6-4732d0911d5a78c5","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","1"],["1","3","1"]],"card_rows":[["1"],["1","1"],["1","3","1"],["1","7","6","1"]],"row_sums_preview":["1","2","5","15","52","203","877","4140"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-6fea5aff4f44fa54","steps":["basis A008277"],"edges":["e-f923f3a2f8b8c5a1"],"text":"A023531 \u2192 basis A008277"},"headline":"A023531 \u2192 basis A008277","chain":null}},{"id":"l-feea80a7b93ef366","kind":"fingerprint_related","node_a_id":"n-cbbf33139cbba3a7","node_b_id":"n-6709415e7c106f96","fingerprint":"f6-4732d0911d5a78c5","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4732d0911d5a78c5"},"other_id":"n-6709415e7c106f96","other":{"id":"n-6709415e7c106f96","name":"basis_map_A008277 of Triangle read by rows: row n consists of n zeros followed by n!.","first_index":0,"fingerprint":"f6-4732d0911d5a78c5","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","1"],["2","6","2"]],"card_rows":[["1"],["1","1"],["2","6","2"],["6","42","36","6"]],"row_sums_preview":["1","2","10","90","1248","24360","631440","20865600"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-fbb4b9430d9de872","steps":["basis A008277"],"edges":["e-65b19db4875af7b0"],"text":"A136572 \u2192 basis A008277"},"headline":"A136572 \u2192 basis A008277","chain":null}},{"id":"l-0b560939cff3969f","kind":"oeis_match","node_a_id":"n-cbbf33139cbba3a7","node_b_id":null,"fingerprint":null,"oeis_id":"A320962","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":0,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":1,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A320962","oeis_nonn":false},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A320962","relation":"match_row_sums"}],"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:      -1,         -1;\n   3:       2,          6,           2;\n   4:      -6,        -42,         -36,           -6;\n   5:      24,        360,         600,          240,           24;\n   6:    -120,      -3720,      -10800,        -7800,        -1800,        -120;\n   7:     720,      45360,      216720,       252000,       100800,       15120,         720;\n   8:   -5040,    -640080,    -4868640,     -8573040,     -5292000,    -1340640,     -141120,      -5040;\n   9:   40320,   10281600,   121968000,    313286400,    280264320,   106686720,    18627840,    1451520,     40320;\n  10: -362880, -185431680, -3385670400, -12376022400, -15431472000, -8283461760, -2133734400, -272160000, -16329600, -362880;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n-1+(n-1)*x)*P(n-1,x) - (n-1)*x*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[-(n - 1 + (n - 1)*x)*P[n - 1, x] - (n - 1)*x*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] = -(n - 1 + (n - 1)*x)*v[i - 1] - (n - 1)*x*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-cf1765de6ac2bd4b","steps":["basis A008277"],"edges":["e-f4658f48c67ff3dd"],"text":"basis-map coefficients of A021012 \u2192 basis A008277"},"headline":"basis-map coefficients of A021012 \u2192 basis A008277","chain":null}