{"id":"n-6cf134ccfa9a5641","revision":1,"first_index":0,"fingerprint":"f6-aaa7f30b59ecd9f9","name":"gamma_polynomial of Square array read by antidiagonals: S(p,q) = (p+q+1)!(2p+2q+1)!/((p+1)!(2p+1)!(q+1)!(2q+1)!) (p,q>=0).","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"],["1"],["1","3"],["1","11"],["1","26","26"],["1","50","170"],["1","85","646","323"],["1","133","1862","3059"],["1","196","4508","16100","4830"],["1","276","9660","62100","60030"],["1","375","18900","195750","404550","80910"],["1","495","34452","534006","1958022","1246014"],["1","638","59334","1305348","7614530","10245004","1463572"],["1","806","97526","2925780","25259234","59703644","26899444"],["1","1001","154154","6111105","74148074","276370094","261184924","27984099"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","1","4","12","53","221","1055","5055","25635","132067","700486","3772990","20688427","114886435","645953452"],"nonn":true,"definitions":[{"position":0,"source":"derived:gamma_polynomial","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["1"]],"recurrence":{"denominator":{"coeffs":[["6"],["7"],["2"]]},"terms":[{"coeff":{"coeffs":[["4","-12","0"],["13","-24","0"],["-2","36","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","6","-24"],["0","18","-72"],["0","0","0"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["2","0","0"],["-6","0","0"],["4","0","0"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":162525,"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-10ee054377433864"]},{"id":162526,"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-10ee054377433864"]},{"id":162527,"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-10ee054377433864"]}],"region_exact":{},"incoming_edges":[{"id":"e-10ee054377433864","operation":{"index_shift":0,"operation":"gamma_polynomial","scale":"1"},"operation_detail":"scale 1, shift 0","conclusion_id":"n-6cf134ccfa9a5641","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":"b275c8d6a5c629467360a41ab2bd881f921a7c8351f41b2500d46615f793b220","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-af1d3c6f634a3212","role":"sequence","name":"Square array read by antidiagonals: S(p,q) = (p+q+1)!(2p+2q+1)!/((p+1)!(2p+1)!(q+1)!(2q+1)!) (p,q>=0).","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":[],"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: 1;\n  2: 1,   3;\n  3: 1,  11;\n  4: 1,  26,   26;\n  5: 1,  50,  170;\n  6: 1,  85,  646,   323;\n  7: 1, 133, 1862,  3059;\n  8: 1, 196, 4508, 16100,  4830;\n  9: 1, 276, 9660, 62100, 60030;\n","recurrence_sentence":"The row polynomials satisfy (2*n^2+7*n+6)*P(n,x) = -(2*n^2-13*n-4-(36*n^2-24*n-12)*x)*P(n-1,x) + ((18*n+6)*x-(72*n+24)*x^2)*d/dx P(n-1,x) + (4*n^2-6*n+2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(2*n^2 - 13*n - 4 - (36*n^2 - 24*n - 12)*x)*P[n - 1, x] + ((18*n + 6)*x - (72*n + 24)*x^2)*D[P[n - 1, x], x] + (4*n^2 - 6*n + 2)*P[n - 2, x])/(2*n^2 + 7*n + 6)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = (-(2*n^2 - 13*n - 4 - (36*n^2 - 24*n - 12)*x)*v[i - 1] + ((18*n + 6)*x - (72*n + 24)*x^2)*deriv(v[i - 1], x) + (4*n^2 - 6*n + 2)*v[i - 2])/(2*n^2 + 7*n + 6)); 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-2245b35ec7cd4857","steps":["\u2299 C(n+2,k+1)","\u03b3"],"edges":["e-00224510dcd0cfb5","e-10ee054377433864"],"text":"A091044 \u2192 \u2299 C(n+2,k+1) \u2192 \u03b3"},"headline":"A091044 \u2192 \u2299 C(n+2,k+1) \u2192 \u03b3","chain":null}
