{"id":"n-393ad1074cdb52c4","revision":0,"first_index":0,"fingerprint":"f6-6dbaa8a13258baa3","name":"theta_plus_one of basis_map_A013609 of Triangle of unsigned Stirling numbers of the first kind (see A048994), read by rows, T(n,k) for 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"],["1","4"],["2","12","12"],["6","44","72","32"],["24","200","420","320","80"],["120","1096","2700","2720","1200","192"],["720","7056","19488","23520","14000","4032","448"],["5040","52272","157584","216608","156800","61824","12544","1024"],["40320","438336","1417488","2153088","1795920","870912","244608","36864","2304"],["362880","4106304","14072400","23157760","21546000","12148416","4233600","890880","103680","5120"],["3628800","42514560","153042912","269104000","273354400","173194560","70682304","18585600","3041280","281600","11264"],["39916800","482175360","1811015712","3368258432","3679658400","2561190720","1181625984","366001152","75271680","9856000","743424","24576"],["479001600","5945771520","23178714624","45248476416","52576546880","39565468800","20155623488","7092873216","1726762752","285542400","30604288","1916928","53248"],["6227020800","79211036160","319160604672","650040099072","796616300480","640534806912","354342532544","138277348352","38406880512","7549301760","1026049024","91693056","4845568","114688"],["87178291200","1133862589440","4705881573888","9951656332800","12777728454400","10879366417920","6455376671744","2745808665600","848820360960","191038848000","30973150208","3522355200","266506240","12042240","245760"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","5","26","154","1044","8028","69264","663696","6999840","80627040","1007441280","13575738240","196287356160","3031488633600","49811492505600"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["1","4"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-1","4","-4"],["2","-4","0"],["-1","0","0"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-1","4"],["2","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":180633,"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-a00024aac65de798"]},{"id":180634,"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-a00024aac65de798"]},{"id":180635,"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-a00024aac65de798"]},{"id":180636,"position":3,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1","region_exact":null,"edges":["e-a00024aac65de798"]}],"region_exact":{},"incoming_edges":[{"id":"e-a00024aac65de798","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-393ad1074cdb52c4","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":"bf19f07fdca947c9577e99c4110eaf869f94748bc6bad3afe84d7bdbba58c81b","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-306852163f00b660","role":"sequence","name":"basis_map_A013609 of Triangle of unsigned Stirling numbers of the first kind (see A048994), read by rows, T(n,k) for 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-a36125e0f7695f8a","kind":"fingerprint_related","node_a_id":"n-393ad1074cdb52c4","node_b_id":"n-6794e86a8babf0ab","fingerprint":"f6-6dbaa8a13258baa3","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-6dbaa8a13258baa3"},"other_id":"n-6794e86a8babf0ab","other":{"id":"n-6794e86a8babf0ab","name":"theta_plus_one of basis_map_A013611 of Triangle of coefficients in expansion of (x+1)*(x+3)*...*(x + 2n - 1) in rising powers of x.","first_index":0,"fingerprint":"f6-6dbaa8a13258baa3","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","8"],["8","48","48"]],"card_rows":[["1"],["2","8"],["8","48","48"],["48","352","576","256"]],"row_sums_preview":["1","10","104","1232","16704","256896","4432896","84953088"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-4f6e747b794bcc2e","steps":["basis A013611","\u03b8+1"],"edges":["e-38e97daa1f393e3a","e-ca3d009605e34346"],"text":"A028338 \u2192 basis A013611 \u2192 \u03b8+1"},"headline":"A028338 \u2192 basis A013611 \u2192 \u03b8+1","chain":null}},{"id":"l-a582edd047eb3309","kind":"fingerprint_related","node_a_id":"n-142cb87fd684dcb1","node_b_id":"n-393ad1074cdb52c4","fingerprint":"f6-6dbaa8a13258baa3","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-6dbaa8a13258baa3"},"other_id":"n-142cb87fd684dcb1","other":{"id":"n-142cb87fd684dcb1","name":"derivative of Coefficients of raising factorial polynomials, T(n,k) = [x^k] p(x, n) where p(x, n) = (m*x + n - 1)*p(x, n - 1) with p[x, 0] = 1, p[x, -1] = 0, p[x, 1] = m*x and m = 2. Triangle read by rows, for n >= 0 and 0 <= k <= n.","first_index":0,"fingerprint":"f6-6dbaa8a13258baa3","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["2"],["2","8"]],"card_rows":[["2"],["2","8"],["4","24","24"],["12","88","144","64"]],"row_sums_preview":["0","2","10","52","308","2088","16056","138528"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-60c7e36cbe887b7e","steps":["D"],"edges":["e-38fcab8fa0d2021e"],"text":"A137320 \u2192 D"},"headline":"A137320 \u2192 D","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:      1,       4;\n  2:      2,      12,       12;\n  3:      6,      44,       72,       32;\n  4:     24,     200,      420,      320,       80;\n  5:    120,    1096,     2700,     2720,     1200,      192;\n  6:    720,    7056,    19488,    23520,    14000,     4032,     448;\n  7:   5040,   52272,   157584,   216608,   156800,    61824,   12544,   1024;\n  8:  40320,  438336,  1417488,  2153088,  1795920,   870912,  244608,  36864,   2304;\n  9: 362880, 4106304, 14072400, 23157760, 21546000, 12148416, 4233600, 890880, 103680, 5120;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n^2-2*n+1+(4*n-4)*x+4*x^2)*P(n-2,x) + (2*n-1+4*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 4*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 4*x;\nP[n_, x_] := P[n, x] = Expand[-(n^2 - 2*n + 1 + (4*n - 4)*x + 4*x^2)*P[n - 2, x] + (2*n - 1 + 4*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = -(n^2 - 2*n + 1 + (4*n - 4)*x + 4*x^2)*v[i - 2] + (2*n - 1 + 4*x)*v[i - 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-b9fbfb57d9269e2f","steps":["basis A013609","\u03b8+1"],"edges":["e-5d127e92ec7ea44a","e-a00024aac65de798"],"text":"A132393 \u2192 basis A013609 \u2192 \u03b8+1"},"headline":"A132393 \u2192 basis A013609 \u2192 \u03b8+1","chain":null}