{"id":"n-1909a84aff4b613a","revision":0,"first_index":0,"fingerprint":"f6-fd15deff59829444","name":"theta_plus_one of Triangle read by rows, T(n,k) = (k+1)*(n+1)!*(n+k)!/((k+1)!^2*(n-k)!) with n >= 0 and 0 <= k <= n.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["2","4"],["6","36","36"],["24","288","720","480"],["120","2400","10800","16800","8400"],["720","21600","151200","403200","453600","181440"],["5040","211680","2116800","8467200","15876000","13970880","4656960"],["40320","2257920","30481920","169344000","465696000","670602240","484323840","138378240"],["362880","26127360","457228800","3353011200","12573792000","26153487360","30512401920","18681062400","4670265600"],["3628800","326592000","7185024000","67060224000","326918592000","915372057600","1525620096000","1494484992000","793945152000","176432256000"],["39916800","4390848000","118552896000","1369944576000","8390910528000","30207277900800","67127284224000","93156231168000","78600570048000","36874341504000","7374868300800"],["479001600","63228211200","2054916864000","28768836096000","215766270720000","966632892825600","2738793196339200","5030436483072000","5973643323648000","4424920980480000","1858466811801600","337903056691200"],["6227020800","971415244800","37399486924800","623324782080000","5609923038720000","30517981330636800","106812934657228800","248503562263756800","388286816037120000","402667809223680000","265760754087628800","101033013950668800","16838835658444800"],["87178291200","15866448998400","713990204928000","13962475118592000","148351298135040000","961316411915059200","4058891516974694400","11596832905641984000","22831264782982656000","31005421310223360000","28524987605405491200","16973546343712358400","5893592480455680000","906706535454720000"],["1307674368000","274611617280000","14279804098560000","323675559567360000","4005485049646080000","30441686377310208000","152208431886551040000","521857480753889280000","1255719563064046080000","2139374070405411840000","2567248884486494208000","2121693292964044800000","1149250533688857600000","367216146859161600000","52459449551308800000"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","6","78","1512","38520","1211760","45304560","1961124480","96427739520","5307348614400","323224411910400","21577450475750400","1566693332159155200","122915603609851852800","10361813256369679104000"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["0"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","0"],["1","2"],["1","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","2","2"],["0","2","2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":190863,"position":0,"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-fb7fdf85ff93da4e"]},{"id":190864,"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-fb7fdf85ff93da4e"]},{"id":190865,"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-fb7fdf85ff93da4e"]}],"region_exact":{},"incoming_edges":[{"id":"e-fb7fdf85ff93da4e","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"euler_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-1909a84aff4b613a","premise_count":1,"theorem":"Laguerre's theorem on multiplier sequences: if f is a real polynomial whose roots are all real and <= 0, then (f(k))_(k >= 0) is a multiplier sequence, so x^k -> f(k) x^k maps every real-rooted polynomial to a real-rooted polynomial or zero (Laguerre; see T. Craven and G. Csordas, Multiplier sequences for fields, Illinois J. Math. 21 (1977); Polya--Schur, Theorem 40 of P. Branden, arXiv:1410.6601: sum_k f(k) x^k / k! = f(theta) e^x = C e^x x^m prod (1 + x/a_j), a_j > 0); f(k) has one sign for k >= 0, so coefficients of one sign stay of one sign; applied to f(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.isMultiplierSequence_eval_of_roots_nonpos","RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos","RealRooted.isMultiplierSequence_iff_preserves"],"certificate_hash":"9a5b94ed1d742beb0ba113b29265a8835c2490ca22a70d19e8ed4e0ef033e103","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-694017a14c5c1f2e","role":"sequence","name":"Triangle read by rows, T(n,k) = (k+1)*(n+1)!*(n+k)!/((k+1)!^2*(n-k)!) with n >= 0 and 0 <= k <= n.","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-11d1aca0905334d8","kind":"fingerprint_related","node_a_id":"n-4689fb1386d440ad","node_b_id":"n-1909a84aff4b613a","fingerprint":"f6-fd15deff59829444","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-fd15deff59829444"},"other_id":"n-4689fb1386d440ad","other":{"id":"n-4689fb1386d440ad","name":"T(n,k) = binomial(n,k)*binomial(n+k,k), 0 <= k <= n, triangle read by rows.","first_index":1,"fingerprint":"f6-fd15deff59829444","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","2"],["1","6","6"]],"card_rows":[["1"],["1","2"],["1","6","6"],["1","12","30","20"]],"row_sums_preview":["1","3","13","63","321","1683","8989","48639"],"regions":["(-inf, +inf)","(-inf, 0)","(-1, 0)","(-inf, 0]"],"region_exact":{},"oeis_ids":["A063007"],"derivation":{"root":"n-4689fb1386d440ad","steps":[],"edges":[],"text":"A063007"},"headline":"A063007: T(n,k) = binomial(n,k)*binomial(n+k,k), 0 <= k <= n, triangle read by rows.","chain":null}},{"id":"l-209082b61838df0c","kind":"fingerprint_related","node_a_id":"n-57c367574a30fd7b","node_b_id":"n-1909a84aff4b613a","fingerprint":"f6-fd15deff59829444","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-fd15deff59829444"},"other_id":"n-57c367574a30fd7b","other":{"id":"n-57c367574a30fd7b","name":"basis_map_A007318 of Square the entries of Pascal's triangle.","first_index":0,"fingerprint":"f6-fd15deff59829444","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","1"],["6","6","1"]],"card_rows":[["1"],["2","1"],["6","6","1"],["20","30","12","1"]],"row_sums_preview":["1","3","13","63","321","1683","8989","48639"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-b13531ad79e7c629","steps":["x \u21a6 x+1"],"edges":["e-041c07f9bf4043d7"],"text":"A008459 \u2192 x \u21a6 x+1"},"headline":"A008459 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-3ded6fda2114e7af","kind":"fingerprint_related","node_a_id":"n-78ba05b4a3dd925d","node_b_id":"n-1909a84aff4b613a","fingerprint":"f6-fd15deff59829444","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-fd15deff59829444"},"other_id":"n-78ba05b4a3dd925d","other":{"id":"n-78ba05b4a3dd925d","name":"Triangle read by rows: T(n,k) is the number of lattice paths from (0,0) to (n,n) using steps E=(1,0), N=(0,1) and D=(1,1) (i.e., bilateral Schroeder paths), having k D=(1,1) steps.","first_index":1,"fingerprint":"f6-fd15deff59829444","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","1"],["6","6","1"]],"card_rows":[["1"],["2","1"],["6","6","1"],["20","30","12","1"]],"row_sums_preview":["1","3","13","63","321","1683","8989","48639"],"regions":["(-inf, +inf)","(-inf, 0)","(-inf, 0]"],"region_exact":{},"oeis_ids":["A104684"],"derivation":{"root":"n-78ba05b4a3dd925d","steps":[],"edges":[],"text":"A104684"},"headline":"A104684: Triangle read by rows: T(n,k) is the number of lattice paths from (0,0) to (n,n) using steps E=(1,0), N=(0,1) and D=(1,1) (i.e., bilateral Schroeder paths), having k D=(1,1) steps.","chain":null}},{"id":"l-425b58ca61767a51","kind":"fingerprint_related","node_a_id":"n-26d4ef3069461915","node_b_id":"n-1909a84aff4b613a","fingerprint":"f6-fd15deff59829444","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-fd15deff59829444"},"other_id":"n-26d4ef3069461915","other":{"id":"n-26d4ef3069461915","name":"reversal of basis_map_A007318 of Square the entries of Pascal's triangle.","first_index":0,"fingerprint":"f6-fd15deff59829444","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","2"],["1","6","6"]],"card_rows":[["1"],["1","2"],["1","6","6"],["1","12","30","20"]],"row_sums_preview":["1","3","13","63","321","1683","8989","48639"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-b13531ad79e7c629","steps":["x \u21a6 x+1","reverse"],"edges":["e-041c07f9bf4043d7","e-6c64c53b3af990de"],"text":"A008459 \u2192 x \u21a6 x+1 \u2192 reverse"},"headline":"A008459 \u2192 x \u21a6 x+1 \u2192 reverse","chain":null}},{"id":"l-6f2363ee912e6762","kind":"fingerprint_related","node_a_id":"n-55a66d3aeb8cbce1","node_b_id":"n-1909a84aff4b613a","fingerprint":"f6-fd15deff59829444","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-fd15deff59829444"},"other_id":"n-55a66d3aeb8cbce1","other":{"id":"n-55a66d3aeb8cbce1","name":"mobius_x_over_x_plus_one of Square the entries of Pascal's triangle.","first_index":0,"fingerprint":"f6-fd15deff59829444","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","2"],["1","6","6"]],"card_rows":[["1"],["1","2"],["1","6","6"],["1","12","30","20"]],"row_sums_preview":["1","3","13","63","321","1683","8989","48639"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-b13531ad79e7c629","steps":["x \u21a6 x/(x+1)"],"edges":["e-2ce1a392fcdf3dd9"],"text":"A008459 \u2192 x \u21a6 x/(x+1)"},"headline":"A008459 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-7b9d9a42941ffeea","kind":"fingerprint_related","node_a_id":"n-e4d248920a317259","node_b_id":"n-1909a84aff4b613a","fingerprint":"f6-fd15deff59829444","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-fd15deff59829444"},"other_id":"n-e4d248920a317259","other":{"id":"n-e4d248920a317259","name":"Triangle read by rows: T(n, k) (0<=k<=2n) is the number of Delannoy paths of length n, having k steps.","first_index":0,"fingerprint":"f6-fd15deff59829444","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["0","1","2"],["0","0","1","6","6"]],"card_rows":[["1"],["0","1","2"],["0","0","1","6","6"],["0","0","0","1","12","30","20"]],"row_sums_preview":["1","3","13","63","321","1683","8989","48639"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":["A109983"],"derivation":{"root":"n-e4d248920a317259","steps":[],"edges":[],"text":"A109983"},"headline":"A109983: Triangle read by rows: T(n, k) (0<=k<=2n) is the number of Delannoy paths of length n, having k steps.","chain":null}},{"id":"l-a847df38705ca436","kind":"fingerprint_related","node_a_id":"n-d50c685731c311f2","node_b_id":"n-1909a84aff4b613a","fingerprint":"f6-fd15deff59829444","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-fd15deff59829444"},"other_id":"n-d50c685731c311f2","other":{"id":"n-d50c685731c311f2","name":"reversal of basis_map_A007318 of Triangle T(n, k) of numbers of square lattice walks that start and end at origin after 2*n steps and contain exactly k steps to the east, possibly touching origin at intermediate stages.","first_index":0,"fingerprint":"f6-fd15deff59829444","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","4"],["6","36","36"]],"card_rows":[["1"],["2","4"],["6","36","36"],["20","240","600","400"]],"row_sums_preview":["1","6","78","1260","22470","424116","8305836","166929048"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-fdc03fad975441d0","steps":["basis A007318","reverse"],"edges":["e-2cd9cd27e716475b","e-9bde97cc0f1e3b64"],"text":"A069466 \u2192 basis A007318 \u2192 reverse"},"headline":"A069466 \u2192 basis A007318 \u2192 reverse","chain":null}},{"id":"l-abfe08cab585c92a","kind":"fingerprint_related","node_a_id":"n-ed80a7592b428c53","node_b_id":"n-1909a84aff4b613a","fingerprint":"f6-fd15deff59829444","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-fd15deff59829444"},"other_id":"n-ed80a7592b428c53","other":{"id":"n-ed80a7592b428c53","name":"basis_map_A007318 of Triangle T(n, k) of numbers of square lattice walks that start and end at origin after 2*n steps and contain exactly k steps to the east, possibly touching origin at intermediate stages.","first_index":0,"fingerprint":"f6-fd15deff59829444","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["4","2"],["36","36","6"]],"card_rows":[["1"],["4","2"],["36","36","6"],["400","600","240","20"]],"row_sums_preview":["1","6","78","1260","22470","424116","8305836","166929048"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-fdc03fad975441d0","steps":["basis A007318"],"edges":["e-2cd9cd27e716475b"],"text":"A069466 \u2192 basis A007318"},"headline":"A069466 \u2192 basis A007318","chain":null}},{"id":"l-cff5b84e1c1e8975","kind":"fingerprint_related","node_a_id":"n-8f0a36c9d23410b3","node_b_id":"n-1909a84aff4b613a","fingerprint":"f6-fd15deff59829444","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-fd15deff59829444"},"other_id":"n-8f0a36c9d23410b3","other":{"id":"n-8f0a36c9d23410b3","name":"mobius_x_over_x_plus_one of Triangle T(n, k) of numbers of square lattice walks that start and end at origin after 2*n steps and contain exactly k steps to the east, possibly touching origin at intermediate stages.","first_index":0,"fingerprint":"f6-fd15deff59829444","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","4"],["6","36","36"]],"card_rows":[["1"],["2","4"],["6","36","36"],["20","240","600","400"]],"row_sums_preview":["1","6","78","1260","22470","424116","8305836","166929048"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-fdc03fad975441d0","steps":["x \u21a6 x/(x+1)"],"edges":["e-e1527edb90b23b46"],"text":"A069466 \u2192 x \u21a6 x/(x+1)"},"headline":"A069466 \u2192 x \u21a6 x/(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:       1;\n  1:       2,         4;\n  2:       6,        36,         36;\n  3:      24,       288,        720,         480;\n  4:     120,      2400,      10800,       16800,         8400;\n  5:     720,     21600,     151200,      403200,       453600,       181440;\n  6:    5040,    211680,    2116800,     8467200,     15876000,     13970880,       4656960;\n  7:   40320,   2257920,   30481920,   169344000,    465696000,    670602240,     484323840,     138378240;\n  8:  362880,  26127360,  457228800,  3353011200,  12573792000,  26153487360,   30512401920,   18681062400,   4670265600;\n  9: 3628800, 326592000, 7185024000, 67060224000, 326918592000, 915372057600, 1525620096000, 1494484992000, 793945152000, 176432256000;\n","recurrence_sentence":"The row polynomials satisfy n*P(n,x) = (n^2+n+(2*n^2+2*n)*x)*P(n-1,x) + ((2*n+2)*x+(2*n+2)*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((n^2 + n + (2*n^2 + 2*n)*x)*P[n - 1, x] + ((2*n + 2)*x + (2*n + 2)*x^2)*D[P[n - 1, x], x])/(n)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((n^2 + n + (2*n^2 + 2*n)*x)*v[i - 1] + ((2*n + 2)*x + (2*n + 2)*x^2)*deriv(v[i - 1], x))/(n)); 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-694017a14c5c1f2e","steps":["\u03b8+1"],"edges":["e-fb7fdf85ff93da4e"],"text":"A253284 \u2192 \u03b8+1"},"headline":"A253284 \u2192 \u03b8+1","chain":null}