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T(n,k) is also the unsigned Stirling number |s(n+1, k+1)|, denoting the number of permutations on n+1 elements that contain exactly k+1 cycles.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"half_affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[[],["1"],["3"],["12","1"],["60","10"],["359","87","1"],["2499","777","21"],["19837","7413","325","1"],["176868","76356","4644","36"],["1750256","850226","65883","874","1"],["19038140","10213610","956505","18370","55"],["225801916","131937146","14411804","365123","1930","1"],["2900457768","1826155188","226850052","7149714","56160","78"],["40116512136","26985990668","3741228205","140934365","1493128","3737","1"],["594454907760","424315993920","64707728085","2830702875","38028900","143955","105"]],"degrees":[-1,0,0,1,1,2,2,3,3,4,4,5,5,6,6],"row_sums":["0","1","3","13","70","447","3297","27576","257904","2667240","30226680","372517920","4960668960","70986162240","1086347505600"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["1"],["3"],["12","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-81","18","-1"],["108","-12","0"],["-54","2","0"],["12","0","0"],["-1","0","0"]]},"deriv_order":0,"offset":4,"sign":"none"},{"coeff":{"coeffs":[["-65","10"],["76","-4"],["-30","0"],["4","0"]]},"deriv_order":0,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["-25","2"],["24","0"],["-6","0"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-6"],["4"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":178266,"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-874245cd54ade2fe"]},{"id":178267,"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-874245cd54ade2fe"]},{"id":178268,"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-874245cd54ade2fe"]}],"region_exact":{},"incoming_edges":[{"id":"e-874245cd54ade2fe","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-3efc720b0ee7ec07","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":"18405e5575653cb8aae6922f065e54183b41883a4fbe71105cab6159f5479e47","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-ead5149323bf4a66","role":"sequence","name":"veronese_section_2_1 of Triangle T(n,k), 0 <= k <= n, read by rows, giving coefficients of the polynomial (x+1)(x+2)...(x+n), expanded in increasing powers of x. T(n,k) is also the unsigned Stirling number |s(n+1, k+1)|, denoting the number of permutations on n+1 elements that contain exactly k+1 cycles.","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":[],"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:       0;\n  1:       1;\n  2:       3;\n  3:      12,      1;\n  4:      60,     10;\n  5:     359,     87,     1;\n  6:    2499,    777,    21;\n  7:   19837,   7413,   325,   1;\n  8:  176868,  76356,  4644,  36;\n  9: 1750256, 850226, 65883, 874, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n^4-12*n^3+54*n^2-108*n+81-(2*n^2-12*n+18)*x+x^2)*P(n-4,x) + (4*n^3-30*n^2+76*n-65-(4*n-10)*x)*P(n-3,x) - (6*n^2-24*n+25-2*x)*P(n-2,x) + (4*n-6)*P(n-1,x) for n >= 4, with P(0,x) = 0, P(1,x) = 1, P(2,x) = 3, P(3,x) = 12 + x.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 1;\nP[2, x_] := 3;\nP[3, x_] := 12 + x;\nP[n_, x_] := P[n, x] = Expand[-(n^4 - 12*n^3 + 54*n^2 - 108*n + 81 - (2*n^2 - 12*n + 18)*x + x^2)*P[n - 4, x] + (4*n^3 - 30*n^2 + 76*n - 65 - (4*n - 10)*x)*P[n - 3, x] - (6*n^2 - 24*n + 25 - 2*x)*P[n - 2, x] + (4*n - 6)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; v[3] = 3; v[4] = 12 + x; for(n = 4, N, my(i = n + 1); v[i] = -(n^4 - 12*n^3 + 54*n^2 - 108*n + 81 - (2*n^2 - 12*n + 18)*x + x^2)*v[i - 4] + (4*n^3 - 30*n^2 + 76*n - 65 - (4*n - 10)*x)*v[i - 3] - (6*n^2 - 24*n + 25 - 2*x)*v[i - 2] + (4*n - 6)*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-efe23a76953cf69d","steps":["Veronese 2 (j=1)","D+1"],"edges":["e-59d75b370e10c587","e-874245cd54ade2fe"],"text":"A130534 \u2192 Veronese 2 (j=1) \u2192 D+1"},"headline":"A130534 \u2192 Veronese 2 (j=1) \u2192 D+1","chain":null}