{"id":"n-e146faf479fcfad1","revision":0,"first_index":0,"fingerprint":"f6-72c8cf9569431b31","name":"derivative of basis_map_A013610 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.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[[],["3"],["15","18"],["78","162","81"],["462","1278","1134","324"],["3132","10440","12555","6480","1215"],["24084","91872","134865","95580","32805","4374"],["207792","879480","1492344","1304100","620865","153090","15309"],["1991088","9162072","17388756","17706276","10478160","3612924","673596","52488"],["20999520","103567248","215116884","246617784","171180135","73850616","19381194","2834352","177147"],["241881120","1265236848","2832338340","3573263160","2807798175","1428605262","471670290","97627680","11514555","590490"],["3022323840","16634128896","39681625896","54208511280","47093314950","27251336574","10660161897","2788687440","467668080","45467730","1948617"],["40727214720","234377618688","590714716680","863437150224","815495011650","523803309282","233961782670","72802063224","15491505150","2149974090","175375530","6377292"],["588862068480","3525649949952","9324705317616","14450978969856","14654819476440","10269028371888","5108776539867","1821383568576","462588035481","81737987760","9550171917","663238368","20726199"],["9094465900800","56417921660160","155736004539024","254063505818304","274013463283560","206792775693504","112573247399613","44836558808184","13085990076159","2768029934670","412987938363","41203683612","2466417681","66961566"]],"degrees":[-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["0","3","33","321","3198","33822","383580","4672980","61065360","853724880","12730525920","201855175200","3393142099200","60298764422400","1129838688115200"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["3"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","0","-9"],["0","-6","0"],["-1","0","0"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["1","6"],["2","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":183661,"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-36508cd2b39dc2d0"]},{"id":183662,"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-36508cd2b39dc2d0"]},{"id":183663,"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-36508cd2b39dc2d0"]},{"id":183664,"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-36508cd2b39dc2d0"]}],"region_exact":{},"incoming_edges":[{"id":"e-36508cd2b39dc2d0","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"derivative_polynomial\",\"coefficients\":[{\"numerator\":[\"0\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-e146faf479fcfad1","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":"be4b1ce6182ba41085cf5da6ee05a3d09de5316bbce3083736089b31f6ca3923","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-c89ee0d57229b178","role":"sequence","name":"basis_map_A013610 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:        3;\n  2:       15,        18;\n  3:       78,       162,        81;\n  4:      462,      1278,      1134,       324;\n  5:     3132,     10440,     12555,      6480,      1215;\n  6:    24084,     91872,    134865,     95580,     32805,     4374;\n  7:   207792,    879480,   1492344,   1304100,    620865,   153090,    15309;\n  8:  1991088,   9162072,  17388756,  17706276,  10478160,  3612924,   673596,   52488;\n  9: 20999520, 103567248, 215116884, 246617784, 171180135, 73850616, 19381194, 2834352, 177147;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n^2+6*n*x+9*x^2)*P(n-2,x) + (2*n+1+6*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 3.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 3;\nP[n_, x_] := P[n, x] = Expand[-(n^2 + 6*n*x + 9*x^2)*P[n - 2, x] + (2*n + 1 + 6*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] = 0; v[2] = 3; for(n = 2, N, my(i = n + 1); v[i] = -(n^2 + 6*n*x + 9*x^2)*v[i - 2] + (2*n + 1 + 6*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-efe23a76953cf69d","steps":["basis A013610","D"],"edges":["e-4deb1a58186d6d0a","e-36508cd2b39dc2d0"],"text":"A130534 \u2192 basis A013610 \u2192 D"},"headline":"A130534 \u2192 basis A013610 \u2192 D","chain":null}