{"id":"n-51d4e271f3dd1e2d","revision":1,"first_index":0,"fingerprint":"f6-0b6450615b0c9b7b","name":"one_plus_derivative 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).","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","1"],["4","7","2"],["7","28","29","6"],["11","80","185","146","24"],["16","185","760","1321","874","120"],["22","371","2380","7231","10444","6084","720"],["29","672","6202","29036","72429","91540","48348","5040"],["37","1128","14154","94332","358869","776028","885212","432144","40320"],["46","1785","29220","262542","1409898","4611405","8932580","9385308","4292496","362880"],["56","2695","55770","649242","4668048","21403635","62283430","110438108","108411336","46916640","3628800"],["67","3916","99935","1462362","13545213","82674768","335309645","888060338","1463519860","1356356376","559628640","39916800"],["79","5512","170027","3053622","35382633","276868020","1487481281","5463724838","13383340828","20729610408","18282431232","7234462080","479001600"],["92","7553","277004","5988983","84822738","826888491","5671638544","27479893441","92955452590","213095234716","312877640712","264229825536","100754029440","6227020800"],["106","10115","434980","11142313","189311122","2247781965","19138588040","117956037199","524379623768","1653879423560","3580603470080","5016870834288","4077210213504","1503961724160","87178291200"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","3","13","70","446","3276","27252","253296","2602224","29288160","358457760","4740577920","67375532160","1024208720640","16583626886400"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["2","1"]],"recurrence":{"denominator":{"coeffs":[["-1"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","0","0"],["-1","2","-1"],["0","-2","2"],["0","0","-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-1","0"],["2","-2"],["0","2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":165360,"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-af9e97c4077fc70b"]},{"id":165361,"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-af9e97c4077fc70b"]},{"id":165362,"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-af9e97c4077fc70b"]},{"id":165363,"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-af9e97c4077fc70b"]}],"region_exact":{},"incoming_edges":[{"id":"e-af9e97c4077fc70b","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"derivative_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-51d4e271f3dd1e2d","premise_count":1,"theorem":"Laguerre's theorem on differential operators: if phi is a real polynomial with only real roots, then phi(D) maps every real-rooted polynomial to a real-rooted polynomial or zero (write phi(D) = C D^j prod (1 + c_i D); D preserves real-rootedness by Rolle's theorem and p + c p' is real-rooted for every real c by Rolle's theorem applied to e^(x/c) p; Polya--Schur 1914); when the coefficients of phi have one sign, coefficients of one sign stay of one sign; applied to phi(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.splits_add_C_mul_derivative_all","RealRooted.splits_C_mul_add_derivative","Polynomial.Splits.derivative"],"certificate_hash":"a68dcba8dfa2d017245aeb3f36a5db0ff54677746acc4a702c2a5625e0b1cc0f","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-4d60c3dd8627c01d","role":"sequence","name":"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).","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-29353e2c12d0b3b3","kind":"oeis_match","node_a_id":"n-51d4e271f3dd1e2d","node_b_id":null,"fingerprint":null,"oeis_id":"A121586","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":1,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A121586","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A121586","relation":"match_row_sums"}],"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,    1;\n  2:  4,    7,     2;\n  3:  7,   28,    29,      6;\n  4: 11,   80,   185,    146,      24;\n  5: 16,  185,   760,   1321,     874,     120;\n  6: 22,  371,  2380,   7231,   10444,    6084,     720;\n  7: 29,  672,  6202,  29036,   72429,   91540,   48348,    5040;\n  8: 37, 1128, 14154,  94332,  358869,  776028,  885212,  432144,   40320;\n  9: 46, 1785, 29220, 262542, 1409898, 4611405, 8932580, 9385308, 4292496, 362880;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = -(n+(2*n^2-2*n)*x+(n^3-2*n^2+n)*x^2)*P(n-2,x) + (2*n-1+(2*n^2-2*n)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n + (2*n^2 - 2*n)*x + (n^3 - 2*n^2 + n)*x^2)*P[n - 2, x] + (2*n - 1 + (2*n^2 - 2*n)*x)*P[n - 1, x])/(n - 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + x; for(n = 2, N, my(i = n + 1); v[i] = (-(n + (2*n^2 - 2*n)*x + (n^3 - 2*n^2 + n)*x^2)*v[i - 2] + (2*n - 1 + (2*n^2 - 2*n)*x)*v[i - 1])/(n - 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-4d60c3dd8627c01d","steps":["D+1"],"edges":["e-af9e97c4077fc70b"],"text":"A094638 \u2192 D+1"},"headline":"A094638 \u2192 D+1","chain":null}
