{"id":"n-5d6fe78c6ecdb1ab","revision":0,"first_index":0,"fingerprint":"f6-3dcce632d1612cce","name":"derivative of Coefficient triangle for polynomials used for o.g.f.s for unsigned Stirling1 diagonals.","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":[[],["1"],["8","2"],["58","44","3"],["444","656","156","4"],["3708","8800","4356","456","5"],["33984","116280","96360","22440","1200","6"],["341136","1570608","1932060","783200","99750","2964","7"],["3733920","22052592","37320192","23062000","5312500","403560","7028","8"],["44339040","324373824","716714712","621429536","223825000","31956960","1531936","16208","9"],["568356480","5014962432","13926491856","16010780352","8241921520","1886218320","176707328","5560304","36648","10"],["7827719040","81576603648","276599370672","404721732096","280206993920","93293764704","14360379432","919011008","19535634","81660","11"],["115336085760","1395858873600","5649238985472","10199461893696","9090943211520","4126320278304","940263941568","101358542208","4565333484","66996960","179916","12"],["1810992556800","25101808035840","119068734167424","258913500559488","287471867322960","169799460396672","53443825045744","8677412099328","675144470052","21898298080","225587868","392856","13"],["30196376985600","473816543086080","2595069759658752","6665697945085824","8985856103452800","6670484835418080","2760559114708640","627131078650496","74775998951712","4298260063400","102217796120","748894680","851552","14"]],"degrees":[-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["0","1","10","105","1260","17325","270270","4729725","91891800","1964187225","45831035250","1159525191825","31623414322500","924984868933125","28887988983603750"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["1"]],"recurrence":{"denominator":{"coeffs":[["-1"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","0","0"],["0","0","-2"],["-1","4","3"],["-1","-2","-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0","0","0"],["0","0","1","-1"],["0","-1","0","1"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-2","2"],["1","-4"],["2","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-1","1"],["0","1","-1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":184435,"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-7f50b453ffc53fb1"]},{"id":184436,"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-7f50b453ffc53fb1"]},{"id":184437,"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-7f50b453ffc53fb1"]},{"id":184438,"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-7f50b453ffc53fb1"]}],"region_exact":{},"incoming_edges":[{"id":"e-7f50b453ffc53fb1","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-5d6fe78c6ecdb1ab","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":"5eebeab792b5bfe95e044f46f4c77e89c6f00f06df7381c5dd22b9f09e55ca75","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-9b757b008ff4762d","role":"sequence","name":"Coefficient triangle for polynomials used for o.g.f.s for unsigned Stirling1 diagonals.","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-56f638c28f6d589b","kind":"fingerprint_related","node_a_id":"n-136521175aa69e42","node_b_id":"n-5d6fe78c6ecdb1ab","fingerprint":"f6-3dcce632d1612cce","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-3dcce632d1612cce"},"other_id":"n-136521175aa69e42","other":{"id":"n-136521175aa69e42","name":"derivative of Triangle related to the o.g.f.s. of the right-hand columns of A130534 (E(x,m=1,n)).","first_index":0,"fingerprint":"f6-3dcce632d1612cce","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],[],["1"]],"card_rows":[["1"],["8","2"],["58","44","3"],["444","656","156","4"]],"row_sums_preview":["0","0","1","10","105","1260","17325","270270"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-e14e37227fef094e","steps":["D"],"edges":["e-fe3065a27dc5d87c"],"text":"A163936 \u2192 D"},"headline":"A163936 \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:        0;\n  1:        1;\n  2:        8,         2;\n  3:       58,        44,         3;\n  4:      444,       656,       156,         4;\n  5:     3708,      8800,      4356,       456,         5;\n  6:    33984,    116280,     96360,     22440,      1200,        6;\n  7:   341136,   1570608,   1932060,    783200,     99750,     2964,       7;\n  8:  3733920,  22052592,  37320192,  23062000,   5312500,   403560,    7028,     8;\n  9: 44339040, 324373824, 716714712, 621429536, 223825000, 31956960, 1531936, 16208, 9;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = -(n^3+n^2+(2*n^3-4*n^2)*x+(n^3-3*n^2+2*n)*x^2)*P(n-2,x) - (n^2*x-n*x^2-(n^2-n)*x^3)*d/dx P(n-2,x) + (2*n^2+n-2+(2*n^2-4*n+2)*x)*P(n-1,x) + ((n-1)*x-(n-1)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^3 + n^2 + (2*n^3 - 4*n^2)*x + (n^3 - 3*n^2 + 2*n)*x^2)*P[n - 2, x] - (n^2*x - n*x^2 - (n^2 - n)*x^3)*D[P[n - 2, x], x] + (2*n^2 + n - 2 + (2*n^2 - 4*n + 2)*x)*P[n - 1, x] + ((n - 1)*x - (n - 1)*x^2)*D[P[n - 1, x], x])/(n - 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = (-(n^3 + n^2 + (2*n^3 - 4*n^2)*x + (n^3 - 3*n^2 + 2*n)*x^2)*v[i - 2] - (n^2*x - n*x^2 - (n^2 - n)*x^3)*deriv(v[i - 2], x) + (2*n^2 + n - 2 + (2*n^2 - 4*n + 2)*x)*v[i - 1] + ((n - 1)*x - (n - 1)*x^2)*deriv(v[i - 1], x))/(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-9b757b008ff4762d","steps":["D"],"edges":["e-7f50b453ffc53fb1"],"text":"A112007 \u2192 D"},"headline":"A112007 \u2192 D","chain":null}
