{"id":"n-af1d3c6f634a3212","revision":2,"first_index":0,"fingerprint":"f6-7a3a2f335019814d","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).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["1","1"],["1","5","1"],["1","14","14","1"],["1","30","84","30","1"],["1","55","330","330","55","1"],["1","91","1001","2145","1001","91","1"],["1","140","2548","10010","10010","2548","140","1"],["1","204","5712","37128","68068","37128","5712","204","1"],["1","285","11628","116280","352716","352716","116280","11628","285","1"],["1","385","21945","319770","1492260","2469012","1492260","319770","21945","385","1"],["1","506","38962","793155","5393454","13728792","13728792","5393454","793155","38962","506","1"],["1","650","65780","1808950","17185025","63740820","98062800","63740820","17185025","1808950","65780","650","1"],["1","819","106470","3848130","49384335","255900645","573667380","573667380","255900645","49384335","3848130","106470","819","1"],["1","1015","166257","7719075","130195065","911365455","2854276425","4159088505","2854276425","911365455","130195065","7719075","166257","1015","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","2","7","30","146","772","4331","25398","154158","961820","6137734","39909740","263665252","1765815560","11966535091"],"nonn":true,"definitions":[{"position":0,"source":"oeis_catalog:A111910","source_kind":"oeis_catalog","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["1","1"]],"recurrence":{"denominator":{"coeffs":[["6"],["7"],["2"]]},"terms":[{"coeff":{"coeffs":[["4","-2"],["13","1"],["-2","16"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","6","-6"],["0","18","-18"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["2","4","2"],["-6","-12","-6"],["4","8","4"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"}},{"position":1,"source":"derived:reversal","source_kind":"derived","equality_edge_id":"e-63f6e30cb237e510","recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["6"],["7"],["2"]]},"terms":[{"coeff":{"coeffs":[["6","0"],["7","-5"],["2","20"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","30","18"],["0","6","-30"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","12","12"]]},"deriv_order":2,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":160242,"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-00224510dcd0cfb5","e-3a2967256f8a0d67"]},{"id":160243,"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-00224510dcd0cfb5","e-3a2967256f8a0d67"]},{"id":160244,"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-00224510dcd0cfb5","e-3a2967256f8a0d67"]}],"region_exact":{},"incoming_edges":[{"id":"e-63f6e30cb237e510","operation":{"operation":"definition_equality"},"operation_detail":null,"conclusion_id":"n-af1d3c6f634a3212","premise_count":0,"theorem":"all-index identity Q_n = P_n certified by exact Ore-algebra division and identifying rows (polytool preserver-image certificate of the identity map)","lean_route":null,"lean_declarations":[],"certificate_hash":"fd42d2c40d75405eca9e665e646eba066f7fea94ce643595ce87f1181f0a7285","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[],"premises":[]},{"id":"e-00224510dcd0cfb5","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"hadamard_multiplier\",\"factor\":{\"terms\":[{\"kind\":\"binomial\",\"index_offset\":2,\"coefficient_offset\":1}]}}"},"operation_detail":"hadamard_multiplier, shift 0","conclusion_id":"n-af1d3c6f634a3212","premise_count":1,"theorem":"Polya--Schur characterization of multiplier sequences (G. Polya and I. Schur, J. Reine Angew. Math. 144 (1914) 89--113; stated as Theorem 40 of P. Branden, Unimodality, log-concavity, real-rootedness and beyond, arXiv:1410.6601): a real sequence (lambda_k) is a multiplier sequence (x^k -> lambda_k x^k maps real-rooted polynomials to real-rooted polynomials or zero) iff sum_k binom(m,k) lambda_k x^k is real-rooted with zeros of one sign for every m, iff sum_k lambda_k x^k / k! (or its reflection) is C x^m e^(a x) prod (1 + alpha_j x) with a, alpha_j >= 0; a product of multiplier sequences is one (composition), and a nonzero factor lambda(n) of the row changes nothing","lean_route":null,"lean_declarations":[],"certificate_hash":"36928ba8d5e14fa1dd3ea9da76a2b1ae420c2ebcf5040559e84a344f649be731","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-2245b35ec7cd4857","role":"sequence","name":"One half of odd-numbered entries of even-numbered rows of Pascal's triangle A007318.","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]"}]},{"id":"e-3a2967256f8a0d67","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"reversal\",\"schedule\":{\"first_index\":0,\"first_degree\":0,\"step\":1}}"},"operation_detail":"reversal, shift 0","conclusion_id":"n-af1d3c6f634a3212","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":"reversed","lean_declarations":["RealRooted.X_pow_mul_reverse_eq_zero_or_splits","RealRooted.reflect_eq_zero_or_splits"],"certificate_hash":"73192a5f5134650a69a73fdfefb68743e417160af90919bf9bc9b9cae415bb9a","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)!) 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v[1] = 1; v[2] = 1 + x; for(n = 2, N, my(i = n + 1); v[i] = (-(2*n^2 - 13*n - 4 - (16*n^2 + n - 2)*x)*v[i - 1] + ((18*n + 6)*x - (18*n + 6)*x^2)*deriv(v[i - 1], x) + (4*n^2 - 6*n + 2 + (8*n^2 - 12*n + 4)*x + (4*n^2 - 6*n + 2)*x^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}},{"oeis_id":"A111910","target":"triangle","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0: 1;\n  1: 1,   1;\n  2: 1,   5,     1;\n  3: 1,  14,    14,      1;\n  4: 1,  30,    84,     30,      1;\n  5: 1,  55,   330,    330,     55,      1;\n  6: 1,  91,  1001,   2145,   1001,     91,      1;\n  7: 1, 140,  2548,  10010,  10010,   2548,    140,     1;\n  8: 1, 204,  5712,  37128,  68068,  37128,   5712,   204,   1;\n  9: 1, 285, 11628, 116280, 352716, 352716, 116280, 11628, 285, 1;\n","recurrence_sentence":"The row polynomials satisfy (2*n^2+7*n+6)*P(n,x) = -(2*n^2-13*n-4-(16*n^2+n-2)*x)*P(n-1,x) + ((18*n+6)*x-(18*n+6)*x^2)*d/dx P(n-1,x) + (4*n^2-6*n+2+(8*n^2-12*n+4)*x+(4*n^2-6*n+2)*x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(2*n^2 - 13*n - 4 - (16*n^2 + n - 2)*x)*P[n - 1, x] + ((18*n + 6)*x - (18*n + 6)*x^2)*D[P[n - 1, x], x] + (4*n^2 - 6*n + 2 + (8*n^2 - 12*n + 4)*x + (4*n^2 - 6*n + 2)*x^2)*P[n - 2, x])/(2*n^2 + 7*n + 6)]];\nTable[PadRight[CoefficientList[P[n, x], x], n + 1], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + x; for(n = 2, N, my(i = n + 1); v[i] = (-(2*n^2 - 13*n - 4 - (16*n^2 + n - 2)*x)*v[i - 1] + ((18*n + 6)*x - (18*n + 6)*x^2)*deriv(v[i - 1], x) + (4*n^2 - 6*n + 2 + (8*n^2 - 12*n + 4)*x + (4*n^2 - 6*n + 2)*x^2)*v[i - 2])/(2*n^2 + 7*n + 6)); v};\nrow(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); 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":true}},{"oeis_id":"A111910","target":"reversed_triangle","first_row":0,"oeis_offset":0,"reversed":true,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0: 1;\n  1: 1,   1;\n  2: 1,   5,     1;\n  3: 1,  14,    14,      1;\n  4: 1,  30,    84,     30,      1;\n  5: 1,  55,   330,    330,     55,      1;\n  6: 1,  91,  1001,   2145,   1001,     91,      1;\n  7: 1, 140,  2548,  10010,  10010,   2548,    140,     1;\n  8: 1, 204,  5712,  37128,  68068,  37128,   5712,   204,   1;\n  9: 1, 285, 11628, 116280, 352716, 352716, 116280, 11628, 285, 1;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, where (2*n^2+7*n+6)*P(n,x) = -(2*n^2-13*n-4-(16*n^2+n-2)*x)*P(n-1,x) + ((18*n+6)*x-(18*n+6)*x^2)*d/dx P(n-1,x) + (4*n^2-6*n+2+(8*n^2-12*n+4)*x+(4*n^2-6*n+2)*x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(2*n^2 - 13*n - 4 - (16*n^2 + n - 2)*x)*P[n - 1, x] + ((18*n + 6)*x - (18*n + 6)*x^2)*D[P[n - 1, x], x] + (4*n^2 - 6*n + 2 + (8*n^2 - 12*n + 4)*x + (4*n^2 - 6*n + 2)*x^2)*P[n - 2, x])/(2*n^2 + 7*n + 6)]];\nTable[Reverse[PadRight[CoefficientList[P[n, x], x], n + 1]], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + x; for(n = 2, N, my(i = n + 1); v[i] = (-(2*n^2 - 13*n - 4 - (16*n^2 + n - 2)*x)*v[i - 1] + ((18*n + 6)*x - (18*n + 6)*x^2)*deriv(v[i - 1], x) + (4*n^2 - 6*n + 2 + (8*n^2 - 12*n + 4)*x + (4*n^2 - 6*n + 2)*x^2)*v[i - 2])/(2*n^2 + 7*n + 6)); v};\nrow(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c = Vecrev(c); 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":true,"trim_zeros":"none","regular_triangle":true}}],"derivation":{"root":"n-2245b35ec7cd4857","steps":["\u2299 C(n+2,k+1)"],"edges":["e-00224510dcd0cfb5"],"text":"A091044 \u2192 \u2299 C(n+2,k+1)"},"headline":"A111910: 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).","chain":"A091044 \u2192 \u2299 C(n+2,k+1)"}