{"id":"n-ed516dafc23c3127","revision":0,"first_index":0,"fingerprint":"f6-7d8c365492dbd6da","name":"theta_plus_one of reversal 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":"irregular","degree_slope":null,"degree_intercept":null,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["1"],["1","2"],["1","6","6"],["1","12","33","24"],["1","20","105","200","120"],["1","30","255","900","1370","720"],["1","42","525","2940","8120","10584","5040"],["1","56","966","7840","33845","78792","91476","40320"],["1","72","1638","18144","112245","403704","826868","876672","362880"],["1","90","2610","37800","316365","1615950","5065760","9381600","9239184","3628800"],["1","110","3960","72600","788865","5412330","23918510","67276000","114782184","106286400","39916800"],["1","132","5775","130680","1787115","15825348","93376745","367965840","947322684","1509179760","1325982240","479001600"],["1","156","8151","223080","3747315","41559804","314931617","1648561200","5914861524","14140148880","21247155072","17837314560","6227020800"],["1","182","11193","364364","7372365","100017918","945255311","6327545224","30025069074","99577037560","223451284056","319160604672","257435867520","87178291200"]],"degrees":[0,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["1","1","3","13","70","446","3276","27252","253296","2602224","29288160","358457760","4740577920","67375532160","1024208720640"],"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"],["1","2"]],"recurrence":{"denominator":{"coeffs":[["-2"],["1"]]},"terms":[{"coeff":{"coeffs":[["1","-4","4"],["-1","6","-8"],["0","-2","5"],["0","0","-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-3","4"],["2","-6"],["0","2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":175137,"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-75b245d54c2ea25e"]},{"id":175138,"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-75b245d54c2ea25e"]},{"id":175139,"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-75b245d54c2ea25e"]},{"id":175140,"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-75b245d54c2ea25e"]}],"region_exact":{},"incoming_edges":[{"id":"e-75b245d54c2ea25e","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-ed516dafc23c3127","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":"b6bd2853f5b5a54198fb0df1a3e896be73a6bcaefb5957458bf72b1c114ac3a2","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-ab4410127fe29869","role":"sequence","name":"reversal 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-dc74b7695f0e9646","kind":"fingerprint_related","node_a_id":"n-ed516dafc23c3127","node_b_id":"n-0d0c19e5a4636085","fingerprint":"f6-7d8c365492dbd6da","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-7d8c365492dbd6da"},"other_id":"n-0d0c19e5a4636085","other":{"id":"n-0d0c19e5a4636085","name":"reversal of theta_plus_one of Triangle read by rows: T(n,k) = |s(n,n+1-k)|, where s(n,k) are the signed Stirling numbers of the first kind A008276 (1 <= k <= n; in other words, the unsigned Stirling numbers of the first kind in reverse order).","first_index":0,"fingerprint":"f6-7d8c365492dbd6da","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"],["24","33","12","1"]],"row_sums_preview":["1","3","13","70","446","3276","27252","253296"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-4d60c3dd8627c01d","steps":["\u03b8+1","reverse"],"edges":["e-8baeeca6d59e95f0","e-6411ace3ea682158"],"text":"A094638 \u2192 \u03b8+1 \u2192 reverse"},"headline":"A094638 \u2192 \u03b8+1 \u2192 reverse","chain":null}},{"id":"l-e61725e6115911ed","kind":"fingerprint_related","node_a_id":"n-994dbf40e5f5e735","node_b_id":"n-ed516dafc23c3127","fingerprint":"f6-7d8c365492dbd6da","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-7d8c365492dbd6da"},"other_id":"n-994dbf40e5f5e735","other":{"id":"n-994dbf40e5f5e735","name":"theta_plus_one of Triangle read by rows: T(n,k) = |s(n,n+1-k)|, where s(n,k) are the signed Stirling numbers of the first kind A008276 (1 <= k <= n; in other words, the unsigned Stirling numbers of the first kind in reverse order).","first_index":0,"fingerprint":"f6-7d8c365492dbd6da","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","33","24"]],"row_sums_preview":["1","3","13","70","446","3276","27252","253296"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-4d60c3dd8627c01d","steps":["\u03b8+1"],"edges":["e-8baeeca6d59e95f0"],"text":"A094638 \u2192 \u03b8+1"},"headline":"A094638 \u2192 \u03b8+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: 1;\n  2: 1,  2;\n  3: 1,  6,    6;\n  4: 1, 12,   33,    24;\n  5: 1, 20,  105,   200,    120;\n  6: 1, 30,  255,   900,   1370,    720;\n  7: 1, 42,  525,  2940,   8120,  10584,   5040;\n  8: 1, 56,  966,  7840,  33845,  78792,  91476,  40320;\n  9: 1, 72, 1638, 18144, 112245, 403704, 826868, 876672, 362880;\n","recurrence_sentence":"The row polynomials satisfy (n-2)*P(n,x) = -(n-1+(2*n^2-6*n+4)*x+(n^3-5*n^2+8*n-4)*x^2)*P(n-2,x) + (2*n-3+(2*n^2-6*n+4)*x)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 1, P(2,x) = 1 + 2*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1;\nP[2, x_] := 1 + 2*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n - 1 + (2*n^2 - 6*n + 4)*x + (n^3 - 5*n^2 + 8*n - 4)*x^2)*P[n - 2, x] + (2*n - 3 + (2*n^2 - 6*n + 4)*x)*P[n - 1, x])/(n - 2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1; v[3] = 1 + 2*x; for(n = 3, N, my(i = n + 1); v[i] = (-(n - 1 + (2*n^2 - 6*n + 4)*x + (n^3 - 5*n^2 + 8*n - 4)*x^2)*v[i - 2] + (2*n - 3 + (2*n^2 - 6*n + 4)*x)*v[i - 1])/(n - 2)); 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":["reverse","\u03b8+1"],"edges":["e-c7203dc00a9aba0b","e-75b245d54c2ea25e"],"text":"A132393 \u2192 reverse \u2192 \u03b8+1"},"headline":"A132393 \u2192 reverse \u2192 \u03b8+1","chain":null}