{"id":"n-7d3228ae4286622e","revision":1,"first_index":0,"fingerprint":"f6-4f01350c3be6484d","name":"gamma_polynomial of Triangle read by rows: T(n, k) = t(n-k, k), where t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), and f(n) = 3*n + 3.","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"],["3"],["9","18"],["27","216"],["81","1782","1296"],["243","12636","33048"],["729","83106","524880","198288"],["2187","524880","6718464","8678016"],["6561","3241134","76212576","225068544","52068096"],["19683","19761732","802594008","4529924352","3481843968"],["59049","119633274","8053338816","78436912464","133766297856","20891063808"],["177147","721342584","78222446496","1231163146944","3888980366976","1981234722816"],["531441","4339747206","742954518000","18059300516160","95269599912960","105663871616256","11887408336896"],["1594323","26076746988","6946975494360","252373423057920","2079178817276160","4188420087003648","1517602034469888"],["4782969","156584839122","64243844750448","3403617753390480","41787366027575040","137766926084350464","107261204289933312","9105612206819328"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","3","27","243","3159","45927","807003","15923547","356596911","8834143743","241267305267","7180322202963","231627475178919","8044547415643287","299389126796441163"],"nonn":true,"definitions":[{"position":0,"source":"derived:gamma_polynomial","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["3","-6","0"],["0","6","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","3","-12"],["0","0","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":162627,"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-184795b7aaceffec"]},{"id":162628,"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-184795b7aaceffec"]},{"id":162629,"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-184795b7aaceffec"]}],"region_exact":{},"incoming_edges":[{"id":"e-184795b7aaceffec","operation":{"index_shift":0,"operation":"gamma_polynomial","scale":"1"},"operation_detail":"scale 1, shift 0","conclusion_id":"n-7d3228ae4286622e","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":"a5e0726d2f236d1f8b1e27dec52dd9c731b1cf983b638eafaf4789e4ab30e0ea","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-dfedc6a36c97eba4","role":"sequence","name":"Triangle read by rows: T(n, k) = t(n-k, k), where t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), and f(n) = 3*n + 3.","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-470de24da77e3302","kind":"fingerprint_related","node_a_id":"n-25862eed72faa70b","node_b_id":"n-7d3228ae4286622e","fingerprint":"f6-4f01350c3be6484d","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4f01350c3be6484d"},"other_id":"n-25862eed72faa70b","other":{"id":"n-25862eed72faa70b","name":"gamma_polynomial of Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 2*x + 2.","first_index":0,"fingerprint":"f6-4f01350c3be6484d","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2"],["4","8"]],"card_rows":[["1"],["2"],["4","8"],["8","64"]],"row_sums_preview":["1","2","12","72","624","6048","70848","931968"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-41809911e05bb4ba","steps":["\u03b3"],"edges":["e-33878cd54e994498"],"text":"A257609 \u2192 \u03b3"},"headline":"A257609 \u2192 \u03b3","chain":null}},{"id":"l-70a452d9f169d3ad","kind":"fingerprint_related","node_a_id":"n-991b9bffcf8ddedb","node_b_id":"n-7d3228ae4286622e","fingerprint":"f6-4f01350c3be6484d","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4f01350c3be6484d"},"other_id":"n-991b9bffcf8ddedb","other":{"id":"n-991b9bffcf8ddedb","name":"Triangle T(n,k) (n >= 1, 0 <= k <= floor((n-1)/2)) read by rows, where T(n,k) = (k+1)T(n-1,k) + (2n-4k)T(n-1,k-1).","first_index":0,"fingerprint":"f6-4f01350c3be6484d","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1"],["1","2"]],"card_rows":[["1"],["1"],["1","2"],["1","8"]],"row_sums_preview":["1","1","3","9","39","189","1107","7281"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":["A101280"],"derivation":{"root":"n-ff270bcb802a85af","steps":["\u03b3 (shift +1)"],"edges":["e-e49ea542bfed6bbf"],"text":"A008292 \u2192 \u03b3 (shift +1)"},"headline":"A101280: Triangle T(n,k) (n >= 1, 0 <= k <= floor((n-1)/2)) read by rows, where T(n,k) = (k+1)T(n-1,k) + (2n-4k)T(n-1,k-1).","chain":"A008292 \u2192 \u03b3 (shift +1)"}},{"id":"l-ba82f233c49b6b62","kind":"fingerprint_related","node_a_id":"n-0942305fd4723770","node_b_id":"n-7d3228ae4286622e","fingerprint":"f6-4f01350c3be6484d","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4f01350c3be6484d"},"other_id":"n-0942305fd4723770","other":{"id":"n-0942305fd4723770","name":"gamma_polynomial of eulerian","first_index":1,"fingerprint":"f6-4f01350c3be6484d","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1"],["1","2"]],"card_rows":[["1"],["1"],["1","2"],["1","8"]],"row_sums_preview":["1","1","3","9","39","189","1107","7281"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ff270bcb802a85af","steps":["\u03b3"],"edges":["e-c5d35f047ff7eb16"],"text":"A008292 \u2192 \u03b3"},"headline":"A008292 \u2192 \u03b3","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:     3;\n  2:     9,       18;\n  3:    27,      216;\n  4:    81,     1782,      1296;\n  5:   243,    12636,     33048;\n  6:   729,    83106,    524880,     198288;\n  7:  2187,   524880,   6718464,    8678016;\n  8:  6561,  3241134,  76212576,  225068544,   52068096;\n  9: 19683, 19761732, 802594008, 4529924352, 3481843968;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (3+(6*n-6)*x)*P(n-1,x) + (3*x-12*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(3 + (6*n - 6)*x)*P[n - 1, x] + (3*x - 12*x^2)*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (3 + (6*n - 6)*x)*v[i - 1] + (3*x - 12*x^2)*deriv(v[i - 1], x)); 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-dfedc6a36c97eba4","steps":["\u03b3"],"edges":["e-184795b7aaceffec"],"text":"A257620 \u2192 \u03b3"},"headline":"A257620 \u2192 \u03b3","chain":null}