{"id":"n-a8a898e8495cff05","revision":1,"first_index":0,"fingerprint":"f6-f68b3fe345bb2270","name":"gamma_polynomial of Triangle read by rows, n>=1, 1<=k<=n, T(n,k) = k*binomial(n,k)^2/(n-k+1).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"half_affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["2"],["3","3"],["4","12"],["5","30","10"],["6","60","60"],["7","105","210","35"],["8","168","560","280"],["9","252","1260","1260","126"],["10","360","2520","4200","1260"],["11","495","4620","11550","6930","462"],["12","660","7920","27720","27720","5544"],["13","858","12870","60060","90090","36036","1716"],["14","1092","20020","120120","252252","168168","24024"],["15","1365","30030","225225","630630","630630","180180","6435"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","2","6","16","45","126","357","1016","2907","8350","24068","69576","201643","585690","1704510"],"nonn":true,"definitions":[{"position":0,"source":"derived:gamma_polynomial","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["2"],["3","3"]],"recurrence":{"denominator":{"coeffs":[["2"],["1"]]},"terms":[{"coeff":{"coeffs":[["3","0"],["3","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","4"],["-3","4"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-1","4"],["1","-4"]]},"deriv_order":0,"offset":3,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":162498,"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-cc6de87a980d771e"]},{"id":162499,"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-cc6de87a980d771e"]},{"id":162500,"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-cc6de87a980d771e"]}],"region_exact":{},"incoming_edges":[{"id":"e-cc6de87a980d771e","operation":{"index_shift":0,"operation":"gamma_polynomial","scale":"1"},"operation_detail":"scale 1, shift 0","conclusion_id":"n-a8a898e8495cff05","premise_count":1,"theorem":"gamma criterion (RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff): if P = sum_j gamma_j x^j (1 + x)^(d - 2j) with deg Gamma <= d/2, then P is nonzero and real-rooted with roots in (-inf, 0] if and only if Gamma(z) = sum_j gamma_j z^j is; Gamma(c z) with c != 0 has the roots of Gamma divided by c","lean_route":null,"lean_declarations":[],"certificate_hash":"0ca3255e6503550df5d6222aede9fc23c8b4c11c6668072b0dafbdffef27aa98","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-984320118b653f6b","role":"sequence","name":"Triangle read by rows, n>=1, 1<=k<=n, T(n,k) = k*binomial(n,k)^2/(n-k+1).","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-34091bd5444f858d","kind":"fingerprint_related","node_a_id":"n-a8a898e8495cff05","node_b_id":"n-61c7ae27dda08fe6","fingerprint":"f6-f68b3fe345bb2270","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f68b3fe345bb2270"},"other_id":"n-61c7ae27dda08fe6","other":{"id":"n-61c7ae27dda08fe6","name":"derivative of Triangle read by rows: T(n,k) is k-th entry of the gamma-vector of the n-dimensional cyclohedron, 0 <= k <= floor(n/2).","first_index":0,"fingerprint":"f6-f68b3fe345bb2270","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],[],["2"]],"card_rows":[["2"],["6"],["12","12"],["20","60"]],"row_sums_preview":["0","0","2","6","24","80","270","882"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-75ccd0453a9b6cec","steps":["\u2299 C(n,k)","D"],"edges":["e-2126fae20f2e5116","e-702433e45b2ced1d"],"text":"A011973 \u2192 \u2299 C(n,k) \u2192 D"},"headline":"A011973 \u2192 \u2299 C(n,k) \u2192 D","chain":null}},{"id":"l-934fdb196d5c629f","kind":"fingerprint_related","node_a_id":"n-169bd483754ab390","node_b_id":"n-a8a898e8495cff05","fingerprint":"f6-f68b3fe345bb2270","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f68b3fe345bb2270"},"other_id":"n-169bd483754ab390","other":{"id":"n-169bd483754ab390","name":"Triangular array of Motzkin polynomial coefficients.","first_index":0,"fingerprint":"f6-f68b3fe345bb2270","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1"],["1","1"]],"card_rows":[["1"],["1"],["1","1"],["1","3"]],"row_sums_preview":["1","1","2","4","9","21","51","127"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":["A055151"],"derivation":{"root":"n-2a05d8340c329b21","steps":["\u03b3 (shift +1)"],"edges":["e-3261655b8c5a1522"],"text":"narayana \u2192 \u03b3 (shift +1)"},"headline":"A055151: Triangular array of Motzkin polynomial coefficients.","chain":"narayana \u2192 \u03b3 (shift +1)"}},{"id":"l-a363ff8d997e7e7b","kind":"fingerprint_related","node_a_id":"n-a8a898e8495cff05","node_b_id":"n-f842b394609a16be","fingerprint":"f6-f68b3fe345bb2270","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f68b3fe345bb2270"},"other_id":"n-f842b394609a16be","other":{"id":"n-f842b394609a16be","name":"derivative of T(n,k) = binomial(n,2*k)*binomial(2*k,k) for 0 <= k <= n, triangle read by rows.","first_index":1,"fingerprint":"f6-f68b3fe345bb2270","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],[],["2"]],"card_rows":[["2"],["6"],["12","12"],["20","60"]],"row_sums_preview":["0","0","2","6","24","80","270","882"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c914b69838ac70ec","steps":["\u2299 1/k! (shift -1)","D"],"edges":["e-4d3032ecd7423900","e-65da073036a5e795"],"text":"A059344 \u2192 \u2299 1/k! (shift -1) \u2192 D"},"headline":"A059344 \u2192 \u2299 1/k! (shift -1) \u2192 D","chain":null}},{"id":"l-d1983b11a97036ae","kind":"fingerprint_related","node_a_id":"n-6bb2422ce01062d8","node_b_id":"n-a8a898e8495cff05","fingerprint":"f6-f68b3fe345bb2270","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f68b3fe345bb2270"},"other_id":"n-6bb2422ce01062d8","other":{"id":"n-6bb2422ce01062d8","name":"gamma_polynomial of narayana","first_index":1,"fingerprint":"f6-f68b3fe345bb2270","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1"],["1","1"]],"card_rows":[["1"],["1"],["1","1"],["1","3"]],"row_sums_preview":["1","1","2","4","9","21","51","127"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-2a05d8340c329b21","steps":["\u03b3"],"edges":["e-dbb6454df235fcfe"],"text":"narayana \u2192 \u03b3"},"headline":"narayana \u2192 \u03b3","chain":null}},{"id":"l-814307f8c70ea41c","kind":"oeis_match","node_a_id":"n-a8a898e8495cff05","node_b_id":null,"fingerprint":null,"oeis_id":"A005717","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":1,"oeis_term_position":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A005717","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A005717","relation":"match_row_sums"}],"bfiles":[{"node_id":"n-a8a898e8495cff05","oeis_id":"A005717","target":"row_sums","oeis_offset":1,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":29,"oeis_terms_available":29,"term_count":200,"raw_size":10332}],"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:  2;\n  2:  3,   3;\n  3:  4,  12;\n  4:  5,  30,   10;\n  5:  6,  60,   60;\n  6:  7, 105,  210,   35;\n  7:  8, 168,  560,  280;\n  8:  9, 252, 1260, 1260,  126;\n  9: 10, 360, 2520, 4200, 1260;\n","recurrence_sentence":"The row polynomials satisfy (n+2)*P(n,x) = (3*n+3)*P(n-1,x) - (3*n-(4*n+4)*x)*P(n-2,x) + (n-1-(4*n-4)*x)*P(n-3,x) for n >= 3, with P(0,x) = 1, P(1,x) = 2, P(2,x) = 3 + 3*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2;\nP[2, x_] := 3 + 3*x;\nP[n_, x_] := P[n, x] = Expand[Together[((3*n + 3)*P[n - 1, x] - (3*n - (4*n + 4)*x)*P[n - 2, x] + (n - 1 - (4*n - 4)*x)*P[n - 3, x])/(n + 2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2; v[3] = 3 + 3*x; for(n = 3, N, my(i = n + 1); v[i] = ((3*n + 3)*v[i - 1] - (3*n - (4*n + 4)*x)*v[i - 2] + (n - 1 - (4*n - 4)*x)*v[i - 3])/(n + 2)); 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-984320118b653f6b","steps":["\u03b3"],"edges":["e-cc6de87a980d771e"],"text":"A132812 \u2192 \u03b3"},"headline":"A132812 \u2192 \u03b3","chain":null}
