{"id":"n-7d59072621b39c7b","revision":0,"first_index":0,"fingerprint":"f6-1db1b1b28f935ec6","name":"derivative of derivative 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":-2,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[[],[],["2"],["12","6"],["70","60","12"],["450","510","180","20"],["3248","4410","2100","420","30"],["26264","40614","23520","6440","840","42"],["236248","403704","269388","90720","16380","1512","56"],["2345400","4342080","3231900","1265460","283500","36540","2520","72"],["25507152","50457000","41003160","18041100","4733190","762300","73920","3960","90"],["301835952","631548456","551948760","266790700","79126740","15011766","1829520","138600","5940","110"],["3863119104","8484089328","7886482032","4121403000","1349706930","290918628","41969928","4015440","244530","8580","132"],["53193434112","121882518576","119492445072","66722375720","23728294590","5671531866","933500568","106162056","8198190","410410","12012","156"],["784313595648","1865935562400","1916659268160","1133267335200","432279687840","112621058550","20631050440","2686483800","247477230","15765750","660660","16380","182"]],"degrees":[-1,-1,0,1,2,3,4,5,6,7,8,9,10,11,12],"row_sums":["0","0","2","18","142","1160","10208","97720","1018008","11507472","140581872","1848236544","26041957632","391738883328","6268657962240"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["0"],["2"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-8","12","-6","1"],["12","-12","3","0"],["-6","3","0","0"],["1","0","0","0"]]},"deriv_order":0,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["-7","9","-3"],["9","-6","0"],["-3","0","0"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-3","3"],["3","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":179023,"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-f988dd810cf0b090"]},{"id":179024,"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-f988dd810cf0b090"]},{"id":179025,"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-f988dd810cf0b090"]}],"region_exact":{},"incoming_edges":[{"id":"e-f988dd810cf0b090","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-7d59072621b39c7b","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":"967c7913413317021cb6d39882585ae773f0f96b87cda223c90c13e291d90918","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-a34e8f08502aec80","role":"sequence","name":"derivative 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":[{"id":"l-6bafb71cb51c94f7","kind":"fingerprint_related","node_a_id":"n-7d59072621b39c7b","node_b_id":"n-96653194bcf0694e","fingerprint":"f6-1db1b1b28f935ec6","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1db1b1b28f935ec6"},"other_id":"n-96653194bcf0694e","other":{"id":"n-96653194bcf0694e","name":"shift_x_plus_one of Triangle related to the asymptotic expansion of E(x,m=3,n).","first_index":0,"fingerprint":"f6-1db1b1b28f935ec6","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["6","3"],["35","30","6"]],"card_rows":[["1"],["6","3"],["35","30","6"],["225","255","90","10"]],"row_sums_preview":["1","9","71","580","5104","48860","509004","5753736"],"regions":["(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-efe23a76953cf69d","steps":["D\u00b2\u2218x (shift +1)","x \u21a6 x+1"],"edges":["e-337faa044bd16e1a","e-caa7a192d916fe49"],"text":"A130534 \u2192 D\u00b2\u2218x (shift +1) \u2192 x \u21a6 x+1"},"headline":"A130534 \u2192 D\u00b2\u2218x (shift +1) \u2192 x \u21a6 x+1","chain":null}},{"id":"l-bad17c255dc300f4","kind":"fingerprint_related","node_a_id":"n-7d59072621b39c7b","node_b_id":"n-fd9c1c0b6460954a","fingerprint":"f6-1db1b1b28f935ec6","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1db1b1b28f935ec6"},"other_id":"n-fd9c1c0b6460954a","other":{"id":"n-fd9c1c0b6460954a","name":"derivative of Triangle of coefficients, read by rows, where the n-th row forms the polynomial P(n,x) = {Sum_{k=1..n} 1/(k+x)}*{Product_{k=1..n} (k+x)}.","first_index":0,"fingerprint":"f6-1db1b1b28f935ec6","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["2"],["12","6"]],"card_rows":[["2"],["12","6"],["70","60","12"],["450","510","180","20"]],"row_sums_preview":["0","2","18","142","1160","10208","97720","1018008"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-714479c216bddb04","steps":["D"],"edges":["e-f02a11d83cde57a8"],"text":"A074246 \u2192 D"},"headline":"A074246 \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:       0;\n  2:       2;\n  3:      12,       6;\n  4:      70,      60,      12;\n  5:     450,     510,     180,      20;\n  6:    3248,    4410,    2100,     420,     30;\n  7:   26264,   40614,   23520,    6440,    840,    42;\n  8:  236248,  403704,  269388,   90720,  16380,  1512,   56;\n  9: 2345400, 4342080, 3231900, 1265460, 283500, 36540, 2520, 72;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (n^3-6*n^2+12*n-8+(3*n^2-12*n+12)*x+(3*n-6)*x^2+x^3)*P(n-3,x) - (3*n^2-9*n+7+(6*n-9)*x+3*x^2)*P(n-2,x) + (3*n-3+3*x)*P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 2.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 0;\nP[2, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[(n^3 - 6*n^2 + 12*n - 8 + (3*n^2 - 12*n + 12)*x + (3*n - 6)*x^2 + x^3)*P[n - 3, x] - (3*n^2 - 9*n + 7 + (6*n - 9)*x + 3*x^2)*P[n - 2, x] + (3*n - 3 + 3*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] = 0; v[3] = 2; for(n = 3, N, my(i = n + 1); v[i] = (n^3 - 6*n^2 + 12*n - 8 + (3*n^2 - 12*n + 12)*x + (3*n - 6)*x^2 + x^3)*v[i - 3] - (3*n^2 - 9*n + 7 + (6*n - 9)*x + 3*x^2)*v[i - 2] + (3*n - 3 + 3*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":["D","D"],"edges":["e-04f1d27fb1fcfee8","e-f988dd810cf0b090"],"text":"A130534 \u2192 D \u2192 D"},"headline":"A130534 \u2192 D \u2192 D","chain":null}
