{"id":"n-96653194bcf0694e","revision":0,"first_index":0,"fingerprint":"f6-1db1b1b28f935ec6","name":"shift_x_plus_one of Triangle related to the asymptotic expansion of E(x,m=3,n).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["6","3"],["35","30","6"],["225","255","90","10"],["1624","2205","1050","210","15"],["13132","20307","11760","3220","420","21"],["118124","201852","134694","45360","8190","756","28"],["1172700","2171040","1615950","632730","141750","18270","1260","36"],["12753576","25228500","20501580","9020550","2366595","381150","36960","1980","45"],["150917976","315774228","275974380","133395350","39563370","7505883","914760","69300","2970","55"],["1931559552","4242044664","3943241016","2060701500","674853465","145459314","20984964","2007720","122265","4290","66"],["26596717056","60941259288","59746222536","33361187860","11864147295","2835765933","466750284","53081028","4099095","205205","6006","78"],["392156797824","932967781200","958329634080","566633667600","216139843920","56310529275","10315525220","1343241900","123738615","7882875","330330","8190","91"],["6165817614720","15170987111472","16240880073600","10096721070800","4092048160200","1147253720613","229813584000","33411446640","3535131600","269479210","14414400","513240","10920","105"],["102992244837120","261233246627712","290196055400544","188615670588800","80617852169400","24084926954088","5206688971484","830057754240","98326413900","8632383760","554005452","25247040","773500","14280","120"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","9","71","580","5104","48860","509004","5753736","70290936","924118272","13020978816","195869441664","3134328981120","53180752331520","953884282141440"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["6","3"],["35","30","6"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","0","0","1"],["0","0","3","0"],["0","3","0","0"],["1","0","0","0"]]},"deriv_order":0,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["-1","-3","-3"],["-3","-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":184106,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, -1]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, -1]","region_exact":null,"edges":["e-caa7a192d916fe49"]},{"id":184107,"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-caa7a192d916fe49"]},{"id":184108,"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-caa7a192d916fe49"]}],"region_exact":{},"incoming_edges":[{"id":"e-caa7a192d916fe49","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-96653194bcf0694e","premise_count":1,"theorem":"a real affine substitution x -> a x + b (a != 0), the reversal x^d p(1/x) (d >= deg p) and a real Mobius map (c x + e)^d p((a x + b)/(c x + e)) (a e - b c != 0, d >= deg p) map real roots to real roots: a root r of p gives the root (r - b)/a, 1/r (and 0) or (e r - b)/(a - c r) (and -e/c); for an input row that is a polynomial the degree condition holds automatically","lean_route":null,"lean_declarations":["Polynomial.Splits.comp_of_degree_le_one"],"certificate_hash":"f4594841e3264150edf9418759a61ae0d932caad9f8507704d8aacf61ad17dff","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}}],"premises":[{"position":0,"node_id":"n-2c8bb380cc75972b","role":"sequence","name":"Triangle related to the asymptotic expansion of E(x,m=3,n).","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-7d59072621b39c7b","other":{"id":"n-7d59072621b39c7b","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.","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"]],"card_rows":[["2"],["12","6"],["70","60","12"],["450","510","180","20"]],"row_sums_preview":["0","0","2","18","142","1160","10208","97720"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"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}},{"id":"l-e56eb44800a6f7ad","kind":"fingerprint_related","node_a_id":"n-fd9c1c0b6460954a","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-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:         1;\n  1:         6,         3;\n  2:        35,        30,         6;\n  3:       225,       255,        90,        10;\n  4:      1624,      2205,      1050,       210,       15;\n  5:     13132,     20307,     11760,      3220,      420,      21;\n  6:    118124,    201852,    134694,     45360,     8190,     756,     28;\n  7:   1172700,   2171040,   1615950,    632730,   141750,   18270,   1260,    36;\n  8:  12753576,  25228500,  20501580,   9020550,  2366595,  381150,  36960,  1980,   45;\n  9: 150917976, 315774228, 275974380, 133395350, 39563370, 7505883, 914760, 69300, 2970, 55;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (n^3+3*n^2*x+3*n*x^2+x^3)*P(n-3,x) - (3*n^2+3*n+1+(6*n+3)*x+3*x^2)*P(n-2,x) + (3*n+3+3*x)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 6 + 3*x, P(2,x) = 35 + 30*x + 6*x^2.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 6 + 3*x;\nP[2, x_] := 35 + 30*x + 6*x^2;\nP[n_, x_] := P[n, x] = Expand[(n^3 + 3*n^2*x + 3*n*x^2 + x^3)*P[n - 3, x] - (3*n^2 + 3*n + 1 + (6*n + 3)*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] = 1; v[2] = 6 + 3*x; v[3] = 35 + 30*x + 6*x^2; for(n = 3, N, my(i = n + 1); v[i] = (n^3 + 3*n^2*x + 3*n*x^2 + x^3)*v[i - 3] - (3*n^2 + 3*n + 1 + (6*n + 3)*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\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}
