{"id":"n-0aa2174fd7ab3229","revision":1,"first_index":1,"fingerprint":"f6-51e3d9694776f525","name":"one_plus_derivative of Triangle read by rows: T(n,k) is the number of permutations of 1..n arranged in a circle with exactly k descents.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"irregular","degree_slope":null,"degree_intercept":null,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["1"],["2","2"],["3","9","3"],["4","36","28","4"],["5","115","220","75","5"],["6","318","1344","1020","186","6"],["7","805","6741","10570","4109","441","7"],["8","1928","29544","86840","66968","15288","1016","8"],["9","4455","118134","600921","843426","372393","54198","2295","9"],["10","10050","443260","3675440","8691840","6855940","1904900","186240","5110","10"],["11","22297","1589863","20554688","77076582","100897258","49463678","9217032","626527","11253","11"],["12","48876","5519364","107599068","610728696","1249003224","1006692552","328392216","42926652","2075964","24564","12"],["13","106171","18705648","535977143","4443140715","13538098470","16901277744","9009781014","2054099307","194615135","6801392","53235","13"],["14","228998","62262984","2570589364","30267175750","132398148162","244045557504","200482016952","74412174522","12300624490","865346104","22090068","114674","14"]],"degrees":[0,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["1","1","4","15","72","420","2880","22680","201600","1995840","21772800","259459200","3353011200","46702656000","697426329600"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"],["1"],["2","2"]],"recurrence":{"denominator":{"coeffs":[["-24","12"],["26","-10"],["-9","2"],["1","0"]]},"terms":[{"coeff":{"coeffs":[["0","0","90","-78","12"],["0","0","-213","161","-22"],["0","0","184","-111","12"],["0","0","-74","31","-2"],["0","0","14","-3","0"],["0","0","-1","0","0"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0","-24","46","-26","4"],["0","0","50","-89","45","-6"],["0","0","-35","55","-22","2"],["0","0","10","-13","3","0"],["0","0","-1","1","0","0"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-12","90","-57","6"],["19","-165","91","-8"],["-8","96","-39","2"],["1","-23","5","0"],["0","2","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-12","18","-6"],["0","19","-27","8"],["0","-8","10","-2"],["0","1","-1","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":164711,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-63c86030c8ae0f8e"]},{"id":164712,"position":1,"claim":"one_signed_coefficients","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":1},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 1: the coefficients of P_n have one sign","region_exact":null,"edges":["e-63c86030c8ae0f8e"]},{"id":164713,"position":2,"claim":"hadamard_kernel","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":1},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-63c86030c8ae0f8e"]},{"id":164714,"position":3,"claim":"interlacing","from_index":2,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2","region_exact":null,"edges":["e-63c86030c8ae0f8e"]}],"region_exact":{},"incoming_edges":[{"id":"e-63c86030c8ae0f8e","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-0aa2174fd7ab3229","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":"1957f73e38e6542438bd003fe7a7ba349f4670d99142dd2fb537f199da31d25e","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-3ffff24fcb4f40a0","role":"sequence","name":"Triangle read by rows: T(n,k) is the number of permutations of 1..n arranged in a circle with exactly k descents.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-8d8ae68903ac7787","kind":"fingerprint_related","node_a_id":"n-47d31470e745e7ae","node_b_id":"n-0aa2174fd7ab3229","fingerprint":"f6-51e3d9694776f525","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-51e3d9694776f525"},"other_id":"n-47d31470e745e7ae","other":{"id":"n-47d31470e745e7ae","name":"one_plus_derivative of Triangle of Eulerian numbers T(n,k), 0 <= k <= n, read by rows.","first_index":0,"fingerprint":"f6-51e3d9694776f525","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","1"],["1","3","1"]],"card_rows":[["1"],["1","1"],["1","3","1"],["1","9","7","1"]],"row_sums_preview":["1","2","5","18","84","480","3240","25200"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-efd1fdf07e5468d8","steps":["reverse (shift +1)","D+1"],"edges":["e-61bc78adfe1eb25a","e-bb4a78f786290f50"],"text":"A173018 \u2192 reverse (shift +1) \u2192 D+1"},"headline":"A173018 \u2192 reverse (shift +1) \u2192 D+1","chain":null}},{"id":"l-7eb04cd25bcf9361","kind":"oeis_match","node_a_id":"n-0aa2174fd7ab3229","node_b_id":null,"fingerprint":null,"oeis_id":"A171005","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":3,"low_zeros_dropped":false,"oeis_offset":3,"oeis_term_position":0,"target":"row_sums","terms_compared":13},"in_catalog":false,"oeis_id":"A171005","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A171005","relation":"match_row_sums"}],"bfiles":[{"node_id":"n-0aa2174fd7ab3229","oeis_id":"A171005","target":"row_sums","oeis_offset":3,"first_row":3,"rows_used":200,"terms_matched":13,"oeis_terms_agreeing":21,"oeis_terms_available":21,"term_count":200,"raw_size":35138}],"oeis_texts":[{"oeis_id":"","target":"node","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1: 1;\n   2: 1;\n   3: 2,    2;\n   4: 3,    9,      3;\n   5: 4,   36,     28,      4;\n   6: 5,  115,    220,     75,      5;\n   7: 6,  318,   1344,   1020,    186,      6;\n   8: 7,  805,   6741,  10570,   4109,    441,     7;\n   9: 8, 1928,  29544,  86840,  66968,  15288,  1016,    8;\n  10: 9, 4455, 118134, 600921, 843426, 372393, 54198, 2295, 9;\n","recurrence_sentence":"The row polynomials satisfy (n^3-9*n^2+26*n-24+(2*n^2-10*n+12)*x)*P(n,x) = -((n^5-14*n^4+74*n^3-184*n^2+213*n-90)*x^2+(3*n^4-31*n^3+111*n^2-161*n+78)*x^3+(2*n^3-12*n^2+22*n-12)*x^4)*P(n-2,x) - ((n^4-10*n^3+35*n^2-50*n+24)*x^2-(n^4-13*n^3+55*n^2-89*n+46)*x^3-(3*n^3-22*n^2+45*n-26)*x^4-(2*n^2-6*n+4)*x^5)*d/dx P(n-2,x) + (n^3-8*n^2+19*n-12+(2*n^4-23*n^3+96*n^2-165*n+90)*x+(5*n^3-39*n^2+91*n-57)*x^2+(2*n^2-8*n+6)*x^3)*P(n-1,x) + ((n^3-8*n^2+19*n-12)*x-(n^3-10*n^2+27*n-18)*x^2-(2*n^2-8*n+6)*x^3)*d/dx P(n-1,x) for n >= 4, with P(1,x) = 1, P(2,x) = 1, P(3,x) = 2 + 2*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 1;\nP[3, x_] := 2 + 2*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-((n^5 - 14*n^4 + 74*n^3 - 184*n^2 + 213*n - 90)*x^2 + (3*n^4 - 31*n^3 + 111*n^2 - 161*n + 78)*x^3 + (2*n^3 - 12*n^2 + 22*n - 12)*x^4)*P[n - 2, x] - ((n^4 - 10*n^3 + 35*n^2 - 50*n + 24)*x^2 - (n^4 - 13*n^3 + 55*n^2 - 89*n + 46)*x^3 - (3*n^3 - 22*n^2 + 45*n - 26)*x^4 - (2*n^2 - 6*n + 4)*x^5)*D[P[n - 2, x], x] + (n^3 - 8*n^2 + 19*n - 12 + (2*n^4 - 23*n^3 + 96*n^2 - 165*n + 90)*x + (5*n^3 - 39*n^2 + 91*n - 57)*x^2 + (2*n^2 - 8*n + 6)*x^3)*P[n - 1, x] + ((n^3 - 8*n^2 + 19*n - 12)*x - (n^3 - 10*n^2 + 27*n - 18)*x^2 - (2*n^2 - 8*n + 6)*x^3)*D[P[n - 1, x], x])/(n^3 - 9*n^2 + 26*n - 24 + (2*n^2 - 10*n + 12)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1; v[3] = 2 + 2*x; for(n = 4, N, my(i = n); v[i] = (-((n^5 - 14*n^4 + 74*n^3 - 184*n^2 + 213*n - 90)*x^2 + (3*n^4 - 31*n^3 + 111*n^2 - 161*n + 78)*x^3 + (2*n^3 - 12*n^2 + 22*n - 12)*x^4)*v[i - 2] - ((n^4 - 10*n^3 + 35*n^2 - 50*n + 24)*x^2 - (n^4 - 13*n^3 + 55*n^2 - 89*n + 46)*x^3 - (3*n^3 - 22*n^2 + 45*n - 26)*x^4 - (2*n^2 - 6*n + 4)*x^5)*deriv(v[i - 2], x) + (n^3 - 8*n^2 + 19*n - 12 + (2*n^4 - 23*n^3 + 96*n^2 - 165*n + 90)*x + (5*n^3 - 39*n^2 + 91*n - 57)*x^2 + (2*n^2 - 8*n + 6)*x^3)*v[i - 1] + ((n^3 - 8*n^2 + 19*n - 12)*x - (n^3 - 10*n^2 + 27*n - 18)*x^2 - (2*n^2 - 8*n + 6)*x^3)*deriv(v[i - 1], x))/(n^3 - 9*n^2 + 26*n - 24 + (2*n^2 - 10*n + 12)*x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-3ffff24fcb4f40a0","steps":["D+1"],"edges":["e-63c86030c8ae0f8e"],"text":"A334218 \u2192 D+1"},"headline":"A334218 \u2192 D+1","chain":null}